Tilings in Art, Math and Science - Bob Culleyluvogt
This talk focuses on plane tilings, how they have historically connected art and mathematics, and more recently have been connected to chemistry. What did the 2011 Nobel Prize in Chemistry have to do with medieval Islamic mosaic patterns? Bob tries to fit these pieces together.
This PDF file is a compilation of different word problems in four areas of Mathematics: Algebra, Trigonometry, Geometry and Statistics. Each Area has two topics and each two topics has two and three word problems with complete solutions.
Elevating Tactical DDD Patterns Through Object CalisthenicsDorra BARTAGUIZ
After immersing yourself in the blue book and its red counterpart, attending DDD-focused conferences, and applying tactical patterns, you're left with a crucial question: How do I ensure my design is effective? Tactical patterns within Domain-Driven Design (DDD) serve as guiding principles for creating clear and manageable domain models. However, achieving success with these patterns requires additional guidance. Interestingly, we've observed that a set of constraints initially designed for training purposes remarkably aligns with effective pattern implementation, offering a more ‘mechanical’ approach. Let's explore together how Object Calisthenics can elevate the design of your tactical DDD patterns, offering concrete help for those venturing into DDD for the first time!
КАТЕРИНА АБЗЯТОВА «Ефективне планування тестування ключові аспекти та практ...QADay
Lviv Direction QADay 2024 (Professional Development)
КАТЕРИНА АБЗЯТОВА
«Ефективне планування тестування ключові аспекти та практичні поради»
https://linktr.ee/qadayua
State of ICS and IoT Cyber Threat Landscape Report 2024 previewPrayukth K V
The IoT and OT threat landscape report has been prepared by the Threat Research Team at Sectrio using data from Sectrio, cyber threat intelligence farming facilities spread across over 85 cities around the world. In addition, Sectrio also runs AI-based advanced threat and payload engagement facilities that serve as sinks to attract and engage sophisticated threat actors, and newer malware including new variants and latent threats that are at an earlier stage of development.
The latest edition of the OT/ICS and IoT security Threat Landscape Report 2024 also covers:
State of global ICS asset and network exposure
Sectoral targets and attacks as well as the cost of ransom
Global APT activity, AI usage, actor and tactic profiles, and implications
Rise in volumes of AI-powered cyberattacks
Major cyber events in 2024
Malware and malicious payload trends
Cyberattack types and targets
Vulnerability exploit attempts on CVEs
Attacks on counties – USA
Expansion of bot farms – how, where, and why
In-depth analysis of the cyber threat landscape across North America, South America, Europe, APAC, and the Middle East
Why are attacks on smart factories rising?
Cyber risk predictions
Axis of attacks – Europe
Systemic attacks in the Middle East
Download the full report from here:
https://sectrio.com/resources/ot-threat-landscape-reports/sectrio-releases-ot-ics-and-iot-security-threat-landscape-report-2024/
Software Delivery At the Speed of AI: Inflectra Invests In AI-Powered QualityInflectra
In this insightful webinar, Inflectra explores how artificial intelligence (AI) is transforming software development and testing. Discover how AI-powered tools are revolutionizing every stage of the software development lifecycle (SDLC), from design and prototyping to testing, deployment, and monitoring.
Learn about:
• The Future of Testing: How AI is shifting testing towards verification, analysis, and higher-level skills, while reducing repetitive tasks.
• Test Automation: How AI-powered test case generation, optimization, and self-healing tests are making testing more efficient and effective.
• Visual Testing: Explore the emerging capabilities of AI in visual testing and how it's set to revolutionize UI verification.
• Inflectra's AI Solutions: See demonstrations of Inflectra's cutting-edge AI tools like the ChatGPT plugin and Azure Open AI platform, designed to streamline your testing process.
Whether you're a developer, tester, or QA professional, this webinar will give you valuable insights into how AI is shaping the future of software delivery.
Kubernetes & AI - Beauty and the Beast !?! @KCD Istanbul 2024Tobias Schneck
As AI technology is pushing into IT I was wondering myself, as an “infrastructure container kubernetes guy”, how get this fancy AI technology get managed from an infrastructure operational view? Is it possible to apply our lovely cloud native principals as well? What benefit’s both technologies could bring to each other?
Let me take this questions and provide you a short journey through existing deployment models and use cases for AI software. On practical examples, we discuss what cloud/on-premise strategy we may need for applying it to our own infrastructure to get it to work from an enterprise perspective. I want to give an overview about infrastructure requirements and technologies, what could be beneficial or limiting your AI use cases in an enterprise environment. An interactive Demo will give you some insides, what approaches I got already working for real.
Slack (or Teams) Automation for Bonterra Impact Management (fka Social Soluti...Jeffrey Haguewood
Sidekick Solutions uses Bonterra Impact Management (fka Social Solutions Apricot) and automation solutions to integrate data for business workflows.
We believe integration and automation are essential to user experience and the promise of efficient work through technology. Automation is the critical ingredient to realizing that full vision. We develop integration products and services for Bonterra Case Management software to support the deployment of automations for a variety of use cases.
This video focuses on the notifications, alerts, and approval requests using Slack for Bonterra Impact Management. The solutions covered in this webinar can also be deployed for Microsoft Teams.
Interested in deploying notification automations for Bonterra Impact Management? Contact us at sales@sidekicksolutionsllc.com to discuss next steps.
UiPath New York Community Day in-person eventDianaGray10
UiPath Community Day is a unique gathering designed to foster collaboration, learning, and networking with automation enthusiasts. Whether you're an automation developer, business analyst, IT professional, solution architect, CoE lead, practitioner or a student/educator excited about the prospects of artificial intelligence and automation technologies in the United States, then the UiPath Community Day is definitely the place you want to be.
Join UiPath leaders, experts from the industry, and the amazing community members and let's connect over expert sessions, demos and use cases around AI in automation as we highlight our technology with a special speaker on Document Understanding.
📌Agenda
3:00 PM Registrations
3:30 PM Welcome note and Introductions | Corina Gheonea (Senior Director of Global UiPath Community)
4:00 PM Introduction to Document Understanding
How to build and deploy Document Understanding process
Where would Document Understanding be used.
Demo
Q&A
4:45 PM Customer/Partner showcase
Accelirate
Intro to Accelirate and history with UiPath
Why are we excited about the new AI features of UiPath?
Customer highlight
a. Document Understanding – BJs Case Study
b. Document Understanding + generative AI
5.30 PM Networking
Le nuove frontiere dell'AI nell'RPA con UiPath Autopilot™UiPathCommunity
In questo evento online gratuito, organizzato dalla Community Italiana di UiPath, potrai esplorare le nuove funzionalità di Autopilot, il tool che integra l'Intelligenza Artificiale nei processi di sviluppo e utilizzo delle Automazioni.
📕 Vedremo insieme alcuni esempi dell'utilizzo di Autopilot in diversi tool della Suite UiPath:
Autopilot per Studio Web
Autopilot per Studio
Autopilot per Apps
Clipboard AI
GenAI applicata alla Document Understanding
👨🏫👨💻 Speakers:
Stefano Negro, UiPath MVPx3, RPA Tech Lead @ BSP Consultant
Flavio Martinelli, UiPath MVP 2023, Technical Account Manager @UiPath
Andrei Tasca, RPA Solutions Team Lead @NTT Data
Key Trends Shaping the Future of Infrastructure.pdfCheryl Hung
Keynote at DIGIT West Expo, Glasgow on 29 May 2024.
Cheryl Hung, ochery.com
Sr Director, Infrastructure Ecosystem, Arm.
The key trends across hardware, cloud and open-source; exploring how these areas are likely to mature and develop over the short and long-term, and then considering how organisations can position themselves to adapt and thrive.
UiPath Test Automation using UiPath Test Suite series, part 3DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 3. In this session, we will cover desktop automation along with UI automation.
Topics covered:
UI automation Introduction,
UI automation Sample
Desktop automation flow
Pradeep Chinnala, Senior Consultant Automation Developer @WonderBotz and UiPath MVP
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
Neuro-symbolic is not enough, we need neuro-*semantic*Frank van Harmelen
Neuro-symbolic (NeSy) AI is on the rise. However, simply machine learning on just any symbolic structure is not sufficient to really harvest the gains of NeSy. These will only be gained when the symbolic structures have an actual semantics. I give an operational definition of semantics as “predictable inference”.
All of this illustrated with link prediction over knowledge graphs, but the argument is general.
Smart TV Buyer Insights Survey 2024 by 91mobiles.pdf91mobiles
91mobiles recently conducted a Smart TV Buyer Insights Survey in which we asked over 3,000 respondents about the TV they own, aspects they look at on a new TV, and their TV buying preferences.
Transcript: Selling digital books in 2024: Insights from industry leaders - T...BookNet Canada
The publishing industry has been selling digital audiobooks and ebooks for over a decade and has found its groove. What’s changed? What has stayed the same? Where do we go from here? Join a group of leading sales peers from across the industry for a conversation about the lessons learned since the popularization of digital books, best practices, digital book supply chain management, and more.
Link to video recording: https://bnctechforum.ca/sessions/selling-digital-books-in-2024-insights-from-industry-leaders/
Presented by BookNet Canada on May 28, 2024, with support from the Department of Canadian Heritage.
The infamous Mallox is the digital Robin Hoods of our time, except they steal from everyone and give to themselves. Since mid-2021, they've been playing hide and seek with unsecured Microsoft SQL servers, encrypting data, and then graciously offering to give it back for a modest Bitcoin donation.
Mallox decided to go shopping for new malware toys, adding the Remcos RAT, BatCloak, and a sprinkle of Metasploit to their collection. They're now playing a game of "Catch me if you can" with antivirus software, using their FUD obfuscator packers to turn their ransomware into the digital equivalent of a ninja.
-------
This document provides a analysis of the Target Company ransomware group, also known as Smallpox, which has been rapidly evolving since its first identification in June 2021.
The analysis delves into various aspects of the group's operations, including its distinctive practice of appending targeted organizations' names to encrypted files, the evolution of its encryption algorithms, and its tactics for establishing persistence and evading defenses.
The insights gained from this analysis are crucial for informing defense strategies and enhancing preparedness against such evolving cyber threats.
2. The following are some key words translated into
mathematical operations.
Linear Word-Problems
3. The following are some key words translated into
mathematical operations.
+: add, sum, plus, total, combine, increased by, # more than ..
Linear Word-Problems
4. The following are some key words translated into
mathematical operations.
+: add, sum, plus, total, combine, increased by, # more than ..
–: subtract, difference, minus, decreased by, # less than ..
Linear Word-Problems
5. The following are some key words translated into
mathematical operations.
+: add, sum, plus, total, combine, increased by, # more than ..
–: subtract, difference, minus, decreased by, # less than ..
*: multiply, product, times, “fractions or %” of the amount ..
Linear Word-Problems
6. The following are some key words translated into
mathematical operations.
+: add, sum, plus, total, combine, increased by, # more than ..
–: subtract, difference, minus, decreased by, # less than ..
*: multiply, product, times, “fractions or %” of the amount ..
Linear Word-Problems
Example A. Let x be an unknown number, write the
mathematical expressions represent the following amounts.
a. 300 more than twice x
7. The following are some key words translated into
mathematical operations.
+: add, sum, plus, total, combine, increased by, # more than ..
–: subtract, difference, minus, decreased by, # less than ..
*: multiply, product, times, “fractions or %” of the amount ..
Linear Word-Problems
Example A. Let x be an unknown number, write the
mathematical expressions represent the following amounts.
a. 300 more than twice x
2x
8. The following are some key words translated into
mathematical operations.
+: add, sum, plus, total, combine, increased by, # more than ..
–: subtract, difference, minus, decreased by, # less than ..
*: multiply, product, times, “fractions or %” of the amount ..
Linear Word-Problems
Example A. Let x be an unknown number, write the
mathematical expressions represent the following amounts.
a. 300 more than twice x
“2x + 300” 2x
9. The following are some key words translated into
mathematical operations.
+: add, sum, plus, total, combine, increased by, # more than ..
–: subtract, difference, minus, decreased by, # less than ..
*: multiply, product, times, “fractions or %” of the amount ..
Linear Word-Problems
Example A. Let x be an unknown number, write the
mathematical expressions represent the following amounts.
a. 300 more than twice x
“2x + 300”
b. 2/3 of x
2x
10. The following are some key words translated into
mathematical operations.
+: add, sum, plus, total, combine, increased by, # more than ..
–: subtract, difference, minus, decreased by, # less than ..
*: multiply, product, times, “fractions or %” of the amount ..
Linear Word-Problems
Example A. Let x be an unknown number, write the
mathematical expressions represent the following amounts.
a. 300 more than twice x
“2x + 300”
b. 2/3 of x
3
“ 2 ”x
2x
11. The following are some key words translated into
mathematical operations.
+: add, sum, plus, total, combine, increased by, # more than ..
–: subtract, difference, minus, decreased by, # less than ..
*: multiply, product, times, “fractions or %” of the amount ..
Linear Word-Problems
Example A. Let x be an unknown number, write the
mathematical expressions represent the following amounts.
a. 300 more than twice x
“2x + 300”
b. 2/3 of x
c. 40% of the sum of x and 100
3
“ 2 ”x
2x
12. The following are some key words translated into
mathematical operations.
+: add, sum, plus, total, combine, increased by, # more than ..
–: subtract, difference, minus, decreased by, # less than ..
*: multiply, product, times, “fractions or %” of the amount ..
Linear Word-Problems
Example A. Let x be an unknown number, write the
mathematical expressions represent the following amounts.
a. 300 more than twice x
“2x + 300”
b. 2/3 of x
c. 40% of the sum of x and 100
3
“ 2 ”x
100
“ 40 ”(x+100)
2x
13. The following are some key words translated into
mathematical operations.
+: add, sum, plus, total, combine, increased by, # more than ..
–: subtract, difference, minus, decreased by, # less than ..
*: multiply, product, times, “fractions or %” of the amount ..
/: divide, quotient, shared equally, ratio ..
Linear Word-Problems
Example A. Let x be an unknown number, write the
mathematical expressions represent the following amounts.
a. 300 more than twice x
“2x + 300”
b. 2/3 of x
c. 40% of the sum of x and 100
3
“ 2 ”x
100
“ 40 ”(x+100)
2x
14. The following are some key words translated into
mathematical operations.
+: add, sum, plus, total, combine, increased by, # more than ..
–: subtract, difference, minus, decreased by, # less than ..
*: multiply, product, times, “fractions or %” of the amount ..
/: divide, quotient, shared equally, ratio ..
Twice = Double = 2* (amount) Square = (amount)2
Linear Word-Problems
Example A. Let x be an unknown number, write the
mathematical expressions represent the following amounts.
a. 300 more than twice x
“2x + 300”
b. 2/3 of x
c. 40% of the sum of x and 100
3
“ 2 ”x
100
“ 40 ”(x+100)
2x
15. The following are some key words translated into
mathematical operations.
+: add, sum, plus, total, combine, increased by, # more than ..
–: subtract, difference, minus, decreased by, # less than ..
*: multiply, product, times, “fractions or %” of the amount ..
/: divide, quotient, shared equally, ratio ..
Twice = Double = 2* (amount) Square = (amount)2
Linear Word-Problems
Example A. Let x be an unknown number, write the
mathematical expressions represent the following amounts.
a. 300 more than twice x
“2x + 300”
b. 2/3 of x
c. 40% of the sum of x and 100
3
“ 2 ”x
100
“ 40 ”(x+100)
e. Divide the difference between
x-square and y-square, by 100.2x
16. The following are some key words translated into
mathematical operations.
+: add, sum, plus, total, combine, increased by, # more than ..
–: subtract, difference, minus, decreased by, # less than ..
*: multiply, product, times, “fractions or %” of the amount ..
/: divide, quotient, shared equally, ratio ..
Twice = Double = 2* (amount) Square = (amount)2
Linear Word-Problems
Example A. Let x be an unknown number, write the
mathematical expressions represent the following amounts.
a. 300 more than twice x
“2x + 300”
b. 2/3 of x
c. 40% of the sum of x and 100
3
“ 2 ”x
100
“ 40 ”(x+100)
e. Divide the difference between
x-square and y-square, by 100.
x2 y2
2x
17. The following are some key words translated into
mathematical operations.
+: add, sum, plus, total, combine, increased by, # more than ..
–: subtract, difference, minus, decreased by, # less than ..
*: multiply, product, times, “fractions or %” of the amount ..
/: divide, quotient, shared equally, ratio ..
Twice = Double = 2* (amount) Square = (amount)2
Linear Word-Problems
Example A. Let x be an unknown number, write the
mathematical expressions represent the following amounts.
a. 300 more than twice x
“2x + 300”
b. 2/3 of x
c. 40% of the sum of x and 100
3
“ 2 ”x
100
“ 40 ”(x+100)
100
(x2 – y2)
e. Divide the difference between
x-square and y-square, by 100.
“ ”
x2 y2
The word "difference“ has two versions:
x–y or y–x. One needs to clarify which
version it’s in question before proceeding.
2x
18. Linear Word-Problems
Example B. Mary and Joe share $10,
Mary gets $9 more than Joe. How much does each get?
Following are the steps to solve word problems.
19. Linear Word-Problems
Example B. Mary and Joe share $10,
Mary gets $9 more than Joe. How much does each get?
Following are the steps to solve word problems.
• Read the whole problem and identify the unknown numbers.
20. Linear Word-Problems
Example B. Mary and Joe share $10,
Mary gets $9 more than Joe. How much does each get?
Following are the steps to solve word problems.
• Read the whole problem and identify the unknown numbers.
There are two unknown numbers.
How much does Mary get?
How much does Joe get?
21. Linear Word-Problems
Example B. Mary and Joe share $10,
Mary gets $9 more than Joe. How much does each get?
Following are the steps to solve word problems.
• Read the whole problem and identify the unknown numbers.
• Select x to represent the unknown number where the other
unknown numbers, if any, are stated in relation to it.
There are two unknown numbers.
How much does Mary get?
How much does Joe get?
22. Following are the steps to solve word problems.
• Read the whole problem and identify the unknown numbers.
• Select x to represent the unknown number where the other
unknown numbers, if any, are stated in relation to it.
Linear Word-Problems
Example B. Mary and Joe share $10,
Mary gets $9 more than Joe. How much does each get?
There are two unknown numbers.
How much does Mary get?
How much does Joe get?
23. Following are the steps to solve word problems.
• Read the whole problem and identify the unknown numbers.
• Select x to represent the unknown number where the other
unknown numbers, if any, are stated in relation to it.
Linear Word-Problems
Example B. Mary and Joe share $10,
Mary gets $9 more than Joe. How much does each get?
There are two unknown numbers.
How much does Mary get?
How much does Joe get?
Mary’s share is given in
term of Jos’s share.
So Joe’s share is x.
24. Following are the steps to solve word problems.
• Read the whole problem and identify the unknown numbers.
• Select x to represent the unknown number where the other
unknown numbers, if any, are stated in relation to it.
Linear Word-Problems
Example B. Mary and Joe share $10,
Mary gets $9 more than Joe. How much does each get?
Let x = $ Joe gets,
There are two unknown numbers.
How much does Mary get?
How much does Joe get?
Mary’s share is given in
term of Jos’s share.
So Joe’s share is x.
25. Following are the steps to solve word problems.
• Read the whole problem and identify the unknown numbers.
• Select x to represent the unknown number where the other
unknown numbers, if any, are stated in relation to it.
• Express the other unknowns in terms of the chosen x.
Linear Word-Problems
Example B. Mary and Joe share $10,
Mary gets $9 more than Joe. How much does each get?
Let x = $ Joe gets,
There are two unknown numbers.
How much does Mary get?
How much does Joe get?
Mary’s share is given in
term of Jos’s share.
So Joe’s share is x.
26. Following are the steps to solve word problems.
• Read the whole problem and identify the unknown numbers.
• Select x to represent the unknown number where the other
unknown numbers, if any, are stated in relation to it.
• Express the other unknowns in terms of the chosen x.
Linear Word-Problems
Example B. Mary and Joe share $10,
Mary gets $9 more than Joe. How much does each get?
Let x = $ Joe gets, so Mary gets (x + 9).
There are two unknown numbers.
How much does Mary get?
How much does Joe get?
Mary’s share is given in
term of Jos’s share.
So Joe’s share is x.
27. Following are the steps to solve word problems.
• Read the whole problem and identify the unknown numbers.
• Select x to represent the unknown number where the other
unknown numbers, if any, are stated in relation to it.
• Express the other unknowns in terms of the chosen x.
• Convert the numerical conclusion stated in the problem,
into an equation-using the above expressions in x’s,
then solve the equation for x .
Linear Word-Problems
Example B. Mary and Joe share $10,
Mary gets $9 more than Joe. How much does each get?
Let x = $ Joe gets, so Mary gets (x + 9).
There are two unknown numbers.
How much does Mary get?
How much does Joe get?
Mary’s share is given in
term of Jos’s share.
So Joe’s share is x.
28. Following are the steps to solve word problems.
• Read the whole problem and identify the unknown numbers.
• Select x to represent the unknown number where the other
unknown numbers, if any, are stated in relation to it.
• Express the other unknowns in terms of the chosen x.
• Convert the numerical conclusion stated in the problem,
into an equation-using the above expressions in x’s,
then solve the equation for x .
Linear Word-Problems
Example B. Mary and Joe share $10,
Mary gets $9 more than Joe. How much does each get?
Let x = $ Joe gets, so Mary gets (x + 9). They’ve $10 in total.
There are two unknown numbers.
How much does Mary get?
How much does Joe get?
Mary’s share is given in
term of Jos’s share.
So Joe’s share is x.
29. Following are the steps to solve word problems.
• Read the whole problem and identify the unknown numbers.
• Select x to represent the unknown number where the other
unknown numbers, if any, are stated in relation to it.
• Express the other unknowns in terms of the chosen x.
• Convert the numerical conclusion stated in the problem,
into an equation-using the above expressions in x’s,
then solve the equation for x .
Linear Word-Problems
Example B. Mary and Joe share $10,
Mary gets $9 more than Joe. How much does each get?
Let x = $ Joe gets, so Mary gets (x + 9). They’ve $10 in total.
x + (x + 9) = 10 There are two unknown numbers.
How much does Mary get?
How much does Joe get?
Mary’s share is given in
term of Jos’s share.
So Joe’s share is x.
30. Following are the steps to solve word problems.
• Read the whole problem and identify the unknown numbers.
• Select x to represent the unknown number where the other
unknown numbers, if any, are stated in relation to it.
• Express the other unknowns in terms of the chosen x.
• Convert the numerical conclusion stated in the problem,
into an equation-using the above expressions in x’s,
then solve the equation for x .
Linear Word-Problems
Example B. Mary and Joe share $10,
Mary gets $9 more than Joe. How much does each get?
Let x = $ Joe gets, so Mary gets (x + 9). They’ve $10 in total.
x + (x + 9) = 10
2x + 9 = 10
2x = 10 – 9
2x = 1
There are two unknown numbers.
How much does Mary get?
How much does Joe get?
Mary’s share is given in
term of Jos’s share.
So Joe’s share is x.
31. Following are the steps to solve word problems.
• Read the whole problem and identify the unknown numbers.
• Select x to represent the unknown number where the other
unknown numbers, if any, are stated in relation to it.
• Express the other unknowns in terms of the chosen x.
• Convert the numerical conclusion stated in the problem,
into an equation-using the above expressions in x’s,
then solve the equation for x .
Linear Word-Problems
Example B. Mary and Joe share $10,
Mary gets $9 more than Joe. How much does each get?
Let x = $ Joe gets, so Mary gets (x + 9). They’ve $10 in total.
x + (x + 9) = 10
2x + 9 = 10
2x = 10 – 9
2x = 1 → x = ½ $ (for Joe)
There are two unknown numbers.
How much does Mary get?
How much does Joe get?
Mary’s share is given in
term of Jos’s share.
So Joe’s share is x.
32. Following are the steps to solve word problems.
• Read the whole problem and identify the unknown numbers.
• Select x to represent the unknown number where the other
unknown numbers, if any, are stated in relation to it.
• Express the other unknowns in terms of the chosen x.
• Convert the numerical conclusion stated in the problem,
into an equation-using the above expressions in x’s,
then solve the equation for x .
Linear Word-Problems
Example B. Mary and Joe share $10,
Mary gets $9 more than Joe. How much does each get?
Let x = $ Joe gets, so Mary gets (x + 9). They’ve $10 in total.
x + (x + 9) = 10
2x + 9 = 10
2x = 10 – 9
2x = 1 → x = ½ $ (for Joe)
and 9 + ½ = 9½ $ (for Mary)
There are two unknown numbers.
How much does Mary get?
How much does Joe get?
Mary’s share is given in
term of Jos’s share.
So Joe’s share is x.
33. Example C.
Mary and Joe are to share $120, Joe gets 2/3 of what Mary
gets. How much does each get?
Linear Word-Problems
34. Example C.
Mary and Joe are to share $120, Joe gets 2/3 of what Mary
gets. How much does each get?
Mary gets $ x,
Linear Word-Problems
35. Example C.
Mary and Joe are to share $120, Joe gets 2/3 of what Mary
gets. How much does each get?
Mary gets $ x, so Joe gets $
3
2 x
Linear Word-Problems
36. Example C.
Mary and Joe are to share $120, Joe gets 2/3 of what Mary
gets. How much does each get?
Mary gets $ x, so Joe gets $
3
2 x
3
2 x + x = 120
Linear Word-Problems
Together they have $120, so
37. Example C.
Mary and Joe are to share $120, Joe gets 2/3 of what Mary
gets. How much does each get?
Mary gets $ x, so Joe gets $
3
2 x
3
2 x + x = 120 multiply both sides by 3
3
2 x + x = 120( ) * 3
Linear Word-Problems
Together they have $120, so
38. Example C.
Mary and Joe are to share $120, Joe gets 2/3 of what Mary
gets. How much does each get?
Mary gets $ x, so Joe gets $
3
2 x
3
2 x + x = 120 multiply both sides by 3
3
2 x + x = 120( ) * 3
Linear Word-Problems
Together they have $120, so
39. Example C.
Mary and Joe are to share $120, Joe gets 2/3 of what Mary
gets. How much does each get?
Mary gets $ x, so Joe gets $
3
2 x
3
2 x + x = 120 multiply both sides by 3
3
2 x + x = 120( ) * 3
3 3
Linear Word-Problems
Together they have $120, so
40. Example C.
Mary and Joe are to share $120, Joe gets 2/3 of what Mary
gets. How much does each get?
Mary gets $ x, so Joe gets $
3
2 x
3
2 x + x = 120 multiply both sides by 3
3
2 x + x = 120( ) * 3
3 3
2x + 3x = 360
Linear Word-Problems
Together they have $120, so
41. Example C.
Mary and Joe are to share $120, Joe gets 2/3 of what Mary
gets. How much does each get?
Mary gets $ x, so Joe gets $
3
2 x
3
2 x + x = 120 multiply both sides by 3
3
2 x + x = 120( ) * 3
3 3
2x + 3x = 360
5x = 360
Linear Word-Problems
Together they have $120, so
42. Example C.
Mary and Joe are to share $120, Joe gets 2/3 of what Mary
gets. How much does each get?
Mary gets $ x, so Joe gets $
3
2 x
3
2 x + x = 120 multiply both sides by 3
3
2 x + x = 120( ) * 3
3 3
2x + 3x = 360
5x = 360
x = 360/5
x = 72
Linear Word-Problems
Together they have $120, so
43. Example C.
Mary and Joe are to share $120, Joe gets 2/3 of what Mary
gets. How much does each get?
Mary gets $ x, so Joe gets $
3
2 x
3
2 x + x = 120 multiply both sides by 3
3
2 x + x = 120( ) * 3
3 3
2x + 3x = 360
5x = 360
x = 360/5
x = 72
Therefore Mary gets $72 and the rest 120 – 72 = $48 is the
amount for Joe.
Linear Word-Problems
Together they have $120, so
44. Making Tables
Linear Word-Problems
If a problem provides the same types of information
for multiple entities, organize the information into a table.
45. Making Tables
Linear Word-Problems
If a problem provides the same types of information
for multiple entities, organize the information into a table.
Example D. Put the following
information into a table.
a. Abe has 20 cats, 30 dogs
and Bob has 25 dogs, 15 cats,
46. Making Tables
Linear Word-Problems
If a problem provides the same types of information
for multiple entities, organize the information into a table.
Example D. Put the following
information into a table.
a. Abe has 20 cats, 30 dogs
and Bob has 25 dogs, 15 cats,
No. of cats No. of dogs
Abe
Bob
47. Making Tables
Linear Word-Problems
If a problem provides the same types of information
for multiple entities, organize the information into a table.
Example D. Put the following
information into a table.
a. Abe has 20 cats, 30 dogs
and Bob has 25 dogs, 15 cats,
No. of cats No. of dogs
Abe 20 30
Bob
48. Making Tables
Linear Word-Problems
If a problem provides the same types of information
for multiple entities, organize the information into a table.
Example D. Put the following
information into a table.
a. Abe has 20 cats, 30 dogs
and Bob has 25 dogs, 15 cats,
No. of cats No. of dogs
Abe 20 30
Bob 15 25
49. Making Tables
Linear Word-Problems
If a problem provides the same types of information
for multiple entities, organize the information into a table.
Example D. Put the following
information into a table.
a. Abe has 20 cats, 30 dogs
and Bob has 25 dogs, 15 cats,
No. of cats No. of dogs
Abe 20 30
Bob 15 25
b. Maria’s grocery list:
6 apples, 4 bananas, 3 cakes.
Don's grocery list:
10 cakes, 2 apples, 6 bananas.
50. Making Tables
Linear Word-Problems
If a problem provides the same types of information
for multiple entities, organize the information into a table.
Example D. Put the following
information into a table.
a. Abe has 20 cats, 30 dogs
and Bob has 25 dogs, 15 cats,
No. of cats No. of dogs
Abe 20 30
Bob 15 25
b. Maria’s grocery list:
6 apples, 4 bananas, 3 cakes.
Don's grocery list:
10 cakes, 2 apples, 6 bananas.
Apple Banana Cake
Maria
Don
51. Making Tables
Linear Word-Problems
If a problem provides the same types of information
for multiple entities, organize the information into a table.
Example D. Put the following
information into a table.
a. Abe has 20 cats, 30 dogs
and Bob has 25 dogs, 15 cats,
No. of cats No. of dogs
Abe 20 30
Bob 15 25
b. Maria’s grocery list:
6 apples, 4 bananas, 3 cakes.
Don's grocery list:
10 cakes, 2 apples, 6 bananas.
Apple Banana Cake
Maria 6 4 3
Don 2 6 10
52. Making Tables
Linear Word-Problems
If a problem provides the same types of information
for multiple entities, organize the information into a table.
Example D. Put the following
information into a table.
a. Abe has 20 cats, 30 dogs
and Bob has 25 dogs, 15 cats,
No. of cats No. of dogs
Abe 20 30
Bob 15 25
b. Maria’s grocery list:
6 apples, 4 bananas, 3 cakes.
Don's grocery list:
10 cakes, 2 apples, 6 bananas.
Apple Banana Cake
Maria 6 4 3
Don 2 6 10
By observing the above, one can easily appreciates the tables
which make the numerical information easier to grasp.
53. Making Tables
Linear Word-Problems
If a problem provides the same types of information
for multiple entities, organize the information into a table.
Example D. Put the following
information into a table.
a. Abe has 20 cats, 30 dogs
and Bob has 25 dogs, 15 cats,
No. of cats No. of dogs
Abe 20 30
Bob 15 25
b. Maria’s grocery list:
6 apples, 4 bananas, 3 cakes.
Don's grocery list:
10 cakes, 2 apples, 6 bananas.
Apple Banana Cake
Maria 6 4 3
Don 2 6 10
By observing the above, one can easily appreciates the tables
which make the numerical information easier to grasp.
Following examples utilize math–formulas of the form A* B = C,
and use tables in the construction of their solutions.
54. Cost Formula
The cost of purchasing multiple units of an item is given here, if
p = price per unit
q = quantity–in number of units
C = Cost of the purchase
Linear Word-Problems
then p*q = C
55. Cost Formula
The cost of purchasing multiple units of an item is given here, if
p = price per unit
q = quantity–in number of units
C = Cost of the purchase
Linear Word-Problems
Example E. a. Peanuts cost $5/lb and cashews cost $7/lb.
We’ve 6 lbs of cashews and 8 lbs of peanuts. Put the data into
a table which
includes the cost
of each item and
the total cost.
then p*q = C
56. Cost Formula
The cost of purchasing multiple units of an item is given here, if
p = price per unit
q = quantity–in number of units
C = Cost of the purchase
Linear Word-Problems
Example E. a. Peanuts cost $5/lb and cashews cost $7/lb.
We’ve 6 lbs of cashews and 8 lbs of peanuts. Put the data into
a table which
includes the cost
of each item and
the total cost.
then p*q = C
price/unit $ quantity of units p*q = Cost $
Peanuts
Cashews
57. Cost Formula
The cost of purchasing multiple units of an item is given here, if
p = price per unit
q = quantity–in number of units
C = Cost of the purchase
Linear Word-Problems
Example E. a. Peanuts cost $5/lb and cashews cost $7/lb.
We’ve 6 lbs of cashews and 8 lbs of peanuts. Put the data into
a table which
includes the cost
of each item and
the total cost.
then p*q = C
price/unit $ quantity of units p*q = Cost $
Peanuts 5 8
Cashews 7 6
58. Cost Formula
The cost of purchasing multiple units of an item is given here, if
p = price per unit
q = quantity–in number of units
C = Cost of the purchase
Linear Word-Problems
Example E. a. Peanuts cost $5/lb and cashews cost $7/lb.
We’ve 6 lbs of cashews and 8 lbs of peanuts. Put the data into
a table which
includes the cost
of each item and
the total cost.
then p*q = C
price/unit $ quantity of units p*q = Cost $
Peanuts 5 8 5*8 = 40
Cashews 7 6 7*6 = 42
59. Cost Formula
The cost of purchasing multiple units of an item is given here, if
p = price per unit
q = quantity–in number of units
C = Cost of the purchase
Linear Word-Problems
Example E. a. Peanuts cost $5/lb and cashews cost $7/lb.
We’ve 6 lbs of cashews and 8 lbs of peanuts. Put the data into
a table which
includes the cost
of each item and
the total cost.
then p*q = C
price/unit $ quantity of units p*q = Cost $
Peanuts 5 8 5*8 = 40
Cashews 7 6 7*6 = 42
TOTAL: 82$
60. Cost Formula
The cost of purchasing multiple units of an item is given here, if
p = price per unit
q = quantity–in number of units
C = Cost of the purchase
Linear Word-Problems
Example E. a. Peanuts cost $5/lb and cashews cost $7/lb.
We’ve 6 lbs of cashews and 8 lbs of peanuts. Put the data into
a table which
includes the cost
of each item and
the total cost.
price/unit $ quantity of units p*q = Cost $
Peanuts 5 8 5*8 = 40
Cashews 7 6 7*6 = 42
b. Make the table if we have x lbs of cashews and 2 more lbs
of peanuts.
TOTAL: 82$
then p*q = C
61. Cost Formula
The cost of purchasing multiple units of an item is given here, if
p = price per unit
q = quantity–in number of units
C = Cost of the purchase
Linear Word-Problems
Example E. a. Peanuts cost $5/lb and cashews cost $7/lb.
We’ve 6 lbs of cashews and 8 lbs of peanuts. Put the data into
a table which
includes the cost
of each item and
the total cost.
price/unit $ quantity of units p*q = Cost $
Peanuts 5 8 5*8 = 40
Cashews 7 6 7*6 = 42
b. Make the table if we have x lbs of cashews and 2 more lbs
of peanuts.
TOTAL: 82$
price/unit $ quantity of units p*q = Cost $
Peanuts
Cashews
then p*q = C
62. Cost Formula
The cost of purchasing multiple units of an item is given here, if
p = price per unit
q = quantity–in number of units
C = Cost of the purchase
Linear Word-Problems
Example E. a. Peanuts cost $5/lb and cashews cost $7/lb.
We’ve 6 lbs of cashews and 8 lbs of peanuts. Put the data into
a table which
includes the cost
of each item and
the total cost.
price/unit $ quantity of units p*q = Cost $
Peanuts 5 8 5*8 = 40
Cashews 7 6 7*6 = 42
b. Make the table if we have x lbs of cashews and 2 more lbs
of peanuts.
TOTAL: 82$
price/unit $ quantity of units p*q = Cost $
Peanuts 5
Cashews 7 x
then p*q = C
63. Cost Formula
The cost of purchasing multiple units of an item is given here, if
p = price per unit
q = quantity–in number of units
C = Cost of the purchase
Linear Word-Problems
Example E. a. Peanuts cost $5/lb and cashews cost $7/lb.
We’ve 6 lbs of cashews and 8 lbs of peanuts. Put the data into
a table which
includes the cost
of each item and
the total cost.
price/unit $ quantity of units p*q = Cost $
Peanuts 5 8 5*8 = 40
Cashews 7 6 7*6 = 42
b. Make the table if we have x lbs of cashews and 2 more lbs
of peanuts.
TOTAL: 82$
price/unit $ quantity of units p*q = Cost $
Peanuts 5 (x + 2)
Cashews 7 x
then p*q = C
64. Cost Formula
The cost of purchasing multiple units of an item is given here, if
p = price per unit
q = quantity–in number of units
C = Cost of the purchase
Linear Word-Problems
Example E. a. Peanuts cost $5/lb and cashews cost $7/lb.
We’ve 6 lbs of cashews and 8 lbs of peanuts. Put the data into
a table which
includes the cost
of each item and
the total cost.
price/unit $ quantity of units p*q = Cost $
Peanuts 5 8 5*8 = 40
Cashews 7 6 7*6 = 42
b. Make the table if we have x lbs of cashews and 2 more lbs
of peanuts.
TOTAL: 82$
price/unit $ quantity of units p*q = Cost $
Peanuts 5 (x + 2) 5(x + 2)
Cashews 7 x 7x
then p*q = C
65. Cost Formula
The cost of purchasing multiple units of an item is given here, if
p = price per unit
q = quantity–in number of units
C = Cost of the purchase
Linear Word-Problems
Example E. a. Peanuts cost $5/lb and cashews cost $7/lb.
We’ve 6 lbs of cashews and 8 lbs of peanuts. Put the data into
a table which
includes the cost
of each item and
the total cost.
price/unit $ quantity of units p*q = Cost $
Peanuts 5 8 5*8 = 40
Cashews 7 6 7*6 = 42
b. Make the table if we have x lbs of cashews and 2 more lbs
of peanuts.
TOTAL: 82$
price/unit $ quantity of units p*q = Cost $
Peanuts 5 (x + 2) 5(x + 2)
Cashews 7 x 7x
TOTAL: 5(x + 2) + 7x
then p*q = C
66. Cost Formula
The cost of purchasing multiple units of an item is given here, if
p = price per unit
q = quantity–in number of units
C = Cost of the purchase
Linear Word-Problems
Example E. a. Peanuts cost $5/lb and cashews cost $7/lb.
We’ve 6 lbs of cashews and 8 lbs of peanuts. Put the data into
a table which
includes the cost
of each item and
the total cost.
price/unit $ quantity of units p*q = Cost $
Peanuts 5 8 5*8 = 40
Cashews 7 6 7*6 = 42
b. Make the table if we have x lbs of cashews and 2 more lbs
of peanuts.
TOTAL: 82$
price/unit $ quantity of units p*q = Cost $
Peanuts 5 (x + 2) 5(x + 2)
Cashews 7 x 7x
TOTAL: 5(x + 2) + 7x → 12x + 10
then p*q = C
67. Linear Word-Problems
Example F. a. Peanuts cost $5/lb and cashews cost $7/lb.
We bought 2 more of lbs of peanuts than cashews
and paid $82 in total. How many lbs of each did we buy?
Now let’s turn the above into a problem.
68. Linear Word-Problems
Example F. a. Peanuts cost $5/lb and cashews cost $7/lb.
We bought 2 more of lbs of peanuts than cashews
and paid $82 in total. How many lbs of each did we buy?
Now let’s turn the above into a problem.
Let x = # of lbs of cashews, then (x + 2) = # of lbs of peanuts.
69. Linear Word-Problems
Example F. a. Peanuts cost $5/lb and cashews cost $7/lb.
We bought 2 more of lbs of peanuts than cashews
and paid $82 in total. How many lbs of each did we buy?
Now let’s turn the above into a problem.
Let x = # of lbs of cashews, then (x + 2) = # of lbs of peanuts.
Organize the information into a table.
70. Linear Word-Problems
Example F. a. Peanuts cost $5/lb and cashews cost $7/lb.
We bought 2 more of lbs of peanuts than cashews
and paid $82 in total. How many lbs of each did we buy?
price/unit $ quantity of units p*q = Cost $
Peanuts 5 (x + 2) 5*(x+2)
Cashews 7 x 7*x
Now let’s turn the above into a problem.
TOTAL: 82$
Let x = # of lbs of cashews, then (x + 2) = # of lbs of peanuts.
Organize the information into a table.
71. Linear Word-Problems
Example F. a. Peanuts cost $5/lb and cashews cost $7/lb.
We bought 2 more of lbs of peanuts than cashews
and paid $82 in total. How many lbs of each did we buy?
price/unit $ quantity of units p*q = Cost $
Peanuts 5 (x + 2) 5*(x+2)
Cashews 7 x 7*x
Now let’s turn the above into a problem.
TOTAL: 82$
Let x = # of lbs of cashews, then (x + 2) = # of lbs of peanuts.
Organize the information into a table.
The total $82 is the sum of itemized costs:
5(x + 2) + 7x = 82
72. Linear Word-Problems
Example F. a. Peanuts cost $5/lb and cashews cost $7/lb.
We bought 2 more of lbs of peanuts than cashews
and paid $82 in total. How many lbs of each did we buy?
price/unit $ quantity of units p*q = Cost $
Peanuts 5 (x + 2) 5*(x+2)
Cashews 7 x 7*x
Now let’s turn the above into a problem.
TOTAL: 82$
Let x = # of lbs of cashews, then (x + 2) = # of lbs of peanuts.
Organize the information into a table.
The total $82 is the sum of itemized costs:
5(x + 2) + 7x = 82
12x + 10 = 82
73. Linear Word-Problems
Example F. a. Peanuts cost $5/lb and cashews cost $7/lb.
We bought 2 more of lbs of peanuts than cashews
and paid $82 in total. How many lbs of each did we buy?
price/unit $ quantity of units p*q = Cost $
Peanuts 5 (x + 2) 5*(x+2)
Cashews 7 x 7*x
Now let’s turn the above into a problem.
TOTAL: 82$
Let x = # of lbs of cashews, then (x + 2) = # of lbs of peanuts.
Organize the information into a table.
The total $82 is the sum of itemized costs:
5(x + 2) + 7x = 82
12x + 10 = 82
12x = 82 – 10
74. Linear Word-Problems
Example F. a. Peanuts cost $5/lb and cashews cost $7/lb.
We bought 2 more of lbs of peanuts than cashews
and paid $82 in total. How many lbs of each did we buy?
price/unit $ quantity of units p*q = Cost $
Peanuts 5 (x + 2) 5*(x+2)
Cashews 7 x 7*x
Now let’s turn the above into a problem.
TOTAL: 82$
Let x = # of lbs of cashews, then (x + 2) = # of lbs of peanuts.
Organize the information into a table.
The total $82 is the sum of itemized costs:
5(x + 2) + 7x = 82
12x + 10 = 82
12x = 82 – 10
12x = 72
x = 72/12 = 6
So there are 6 lb of cashews
and 8 lb of peanuts.
76. Concentration Formula
The concentration of a chemical in a solution is usually
given in %. Let
r = concentration (in %)
S = amount of solution
Linear Word-Problems
77. Concentration Formula
The concentration of a chemical in a solution is usually
given in %. Let
r = concentration (in %)
S = amount of solution
C = amount of chemical
Linear Word-Problems
78. Concentration Formula
The concentration of a chemical in a solution is usually
given in %. Let
r = concentration (in %)
S = amount of solution
C = amount of chemical
Then r*S = C
Linear Word-Problems
79. Concentration Formula
The concentration of a chemical in a solution is usually
given in %. Let
r = concentration (in %)
S = amount of solution
C = amount of chemical
Then r*S = C
Linear Word-Problems
Example G. We have 60 gallons brine (salt water) of 25%
concentration. How much salt is there?
80. Concentration Formula
The concentration of a chemical in a solution is usually
given in %. Let
r = concentration (in %)
S = amount of solution
C = amount of chemical
Then r*S = C
Linear Word-Problems
Example G. We have 60 gallons brine (salt water) of 25%
concentration. How much salt is there?
r = concentration = 25% =
100
25
81. Concentration Formula
The concentration of a chemical in a solution is usually
given in %. Let
r = concentration (in %)
S = amount of solution
C = amount of chemical
Then r*S = C
Linear Word-Problems
Example G. We have 60 gallons brine (salt water) of 25%
concentration. How much salt is there?
r = concentration = 25% =
S = amount of brine = 60 gal
100
25
82. Concentration Formula
The concentration of a chemical in a solution is usually
given in %. Let
r = concentration (in %)
S = amount of solution
C = amount of chemical
Then r*S = C
Linear Word-Problems
Example G. We have 60 gallons brine (salt water) of 25%
concentration. How much salt is there?
r = concentration = 25% =
S = amount of brine = 60 gal
C = amount of salt = ?
100
25
83. Concentration Formula
The concentration of a chemical in a solution is usually
given in %. Let
r = concentration (in %)
S = amount of solution
C = amount of chemical
Then r*S = C
100
25
* 60
Linear Word-Problems
Example G. We have 60 gallons brine (salt water) of 25%
concentration. How much salt is there?
r = concentration = 25% =
S = amount of brine = 60 gal
C = amount of salt = ?
100
25
Using rS =
84. Concentration Formula
The concentration of a chemical in a solution is usually
given in %. Let
r = concentration (in %)
S = amount of solution
C = amount of chemical
Then r*S = C
100
25
* 60
Linear Word-Problems
Example G. We have 60 gallons brine (salt water) of 25%
concentration. How much salt is there?
r = concentration = 25% =
S = amount of brine = 60 gal
C = amount of salt = ?
100
25
Using rS =
= * 60
4
1
85. Concentration Formula
The concentration of a chemical in a solution is usually
given in %. Let
r = concentration (in %)
S = amount of solution
C = amount of chemical
Then r*S = C
100
25
* 60
Linear Word-Problems
Example G. We have 60 gallons brine (salt water) of 25%
concentration. How much salt is there?
r = concentration = 25% =
S = amount of brine = 60 gal
C = amount of salt = ?
100
25
Using rS =
= * 60 =
4
1 15
15 gallons of salt.
86. Example H. (Mixture problem) How many gallons of 10% brine
must be added to 30 gallons of 40% brine to dilute the mixture
to 20% brine?
Linear Word-Problems
87. Example H. (Mixture problem) How many gallons of 10% brine
must be added to 30 gallons of 40% brine to dilute the mixture
to 20% brine?
Make the standard table for mixture problems
concentration # of gallons salt (r*S)
Linear Word-Problems
88. Example H. (Mixture problem) How many gallons of 10% brine
must be added to 30 gallons of 40% brine to dilute the mixture
to 20% brine?
Make the standard table for mixture problems
concentration # of gallons salt (r*S)
10%
40%
20%
Linear Word-Problems
89. Example H. (Mixture problem) How many gallons of 10% brine
must be added to 30 gallons of 40% brine to dilute the mixture
to 20% brine?
Make the standard table for mixture problems
concentration # of gallons salt (r*S)
10% x
40%
20%
Linear Word-Problems
90. Example H. (Mixture problem) How many gallons of 10% brine
must be added to 30 gallons of 40% brine to dilute the mixture
to 20% brine?
Make the standard table for mixture problems
concentration # of gallons salt (r*S)
10% x
40% 30
20%
Linear Word-Problems
91. Example H. (Mixture problem) How many gallons of 10% brine
must be added to 30 gallons of 40% brine to dilute the mixture
to 20% brine?
Make the standard table for mixture problems
concentration # of gallons salt (r*S)
10% x
40% 30
20% (x+30)
Linear Word-Problems
92. Example H. (Mixture problem) How many gallons of 10% brine
must be added to 30 gallons of 40% brine to dilute the mixture
to 20% brine?
Make the standard table for mixture problems
concentration # of gallons salt (r*S)
10% x
40% 30
20% (x+30)
10
100 x
Linear Word-Problems
93. Example H. (Mixture problem) How many gallons of 10% brine
must be added to 30 gallons of 40% brine to dilute the mixture
to 20% brine?
Make the standard table for mixture problems
concentration # of gallons salt (r*S)
10% x
40% 30
20% (x+30)
10
100 x
40
100 * 30
Linear Word-Problems
94. Example H. (Mixture problem) How many gallons of 10% brine
must be added to 30 gallons of 40% brine to dilute the mixture
to 20% brine?
Make the standard table for mixture problems
concentration # of gallons salt (r*S)
10% x
40% 30
20% (x+30)
10
100 x
40
100 * 30
20
100
(x+30)
Total amount
of salt in the final
mixture
Linear Word-Problems
95. Example H. (Mixture problem) How many gallons of 10% brine
must be added to 30 gallons of 40% brine to dilute the mixture
to 20% brine?
Make the standard table for mixture problems
concentration # of gallons salt (r*S)
10% x
40% 30
20% (x+30)
10
100 x
40
100 * 30
20
100
(x+30)
Total amount
of salt in the final
mixture
Hence
10
100x +
40
100* 30 =
20
100 (x + 30)
Linear Word-Problems
96. Example H. (Mixture problem) How many gallons of 10% brine
must be added to 30 gallons of 40% brine to dilute the mixture
to 20% brine?
Make the standard table for mixture problems
concentration # of gallons salt (r*S)
10% x
40% 30
20% (x+30)
10
100 x
40
100 * 30
20
100
(x+30)
Total amount
of salt in the final
mixture
Hence
10
100x +
40
100* 30 =
20
100 (x + 30)
10
100x + 40
100* 30 = 20
100(x + 30)[ ] multiply by 100 to
clear denominator.
* 100
Linear Word-Problems
97. Example H. (Mixture problem) How many gallons of 10% brine
must be added to 30 gallons of 40% brine to dilute the mixture
to 20% brine?
Make the standard table for mixture problems
concentration # of gallons salt (r*S)
10% x
40% 30
20% (x+30)
10
100 x
40
100 * 30
20
100
(x+30)
Total amount
of salt in the final
mixture
Hence
10
100x +
40
100* 30 =
20
100 (x + 30)
10
100x + 40
100* 30 = 20
100(x + 30)[ ] multiply by 100 to
clear denominator.
* 100
Linear Word-Problems
98. Example H. (Mixture problem) How many gallons of 10% brine
must be added to 30 gallons of 40% brine to dilute the mixture
to 20% brine?
Make the standard table for mixture problems
concentration # of gallons salt (r*S)
10% x
40% 30
20% (x+30)
10
100 x
40
100 * 30
20
100
(x+30)
Total amount
of salt in the final
mixture
Hence
10
100x +
40
100* 30 =
20
100 (x + 30)
10
100x + 40
100* 30 = 20
100(x + 30)[ ] multiply by 100 to
clear denominator.
* 100
10x + 1200 = 20(x+30)
Linear Word-Problems
99. Example H. (Mixture problem) How many gallons of 10% brine
must be added to 30 gallons of 40% brine to dilute the mixture
to 20% brine?
Make the standard table for mixture problems
concentration # of gallons salt (r*S)
10% x
40% 30
20% (x+30)
10
100 x
40
100 * 30
20
100
(x+30)
Total amount
of salt in the final
mixture
Hence
10
100x +
40
100* 30 =
20
100 (x + 30)
10
100x + 40
100* 30 = 20
100(x + 30)[ ] multiply by 100 to
clear denominator.
* 100
10x + 1200 = 20(x+30)
10x + 1200 = 20x + 600
Linear Word-Problems
100. Example H. (Mixture problem) How many gallons of 10% brine
must be added to 30 gallons of 40% brine to dilute the mixture
to 20% brine?
Make the standard table for mixture problems
concentration # of gallons salt (r*S)
10% x
40% 30
20% (x+30)
10
100 x
40
100 * 30
20
100
(x+30)
Total amount
of salt in the final
mixture
Hence
10
100x +
40
100* 30 =
20
100 (x + 30)
10
100x + 40
100* 30 = 20
100(x + 30)[ ] multiply by 100 to
clear denominator.
* 100
10x + 1200 = 20(x+30)
10x + 1200 = 20x + 600
–600 + 1200 = 20x –10x
Linear Word-Problems
101. Example H. (Mixture problem) How many gallons of 10% brine
must be added to 30 gallons of 40% brine to dilute the mixture
to 20% brine?
Make the standard table for mixture problems
concentration # of gallons salt (r*S)
10% x
40% 30
20% (x+30)
10
100 x
40
100 * 30
20
100
(x+30)
Total amount
of salt in the final
mixture
Hence
10
100x +
40
100* 30 =
20
100 (x + 30)
10
100x + 40
100* 30 = 20
100(x + 30)[ ] multiply by 100 to
clear denominator.
* 100
10x + 1200 = 20(x+30)
10x + 1200 = 20x + 600
–600 + 1200 = 20x –10x
600 = 10x
Linear Word-Problems
102. Example H. (Mixture problem) How many gallons of 10% brine
must be added to 30 gallons of 40% brine to dilute the mixture
to 20% brine?
Make the standard table for mixture problems
concentration # of gallons salt (r*S)
10% x
40% 30
20% (x+30)
10
100 x
40
100 * 30
20
100
(x+30)
Total amount
of salt in the final
mixture
Hence
10
100x +
40
100* 30 =
20
100 (x + 30)
10
100x + 40
100* 30 = 20
100(x + 30)[ ] multiply by 100 to
clear denominator.
* 100
10x + 1200 = 20(x+30)
10x + 1200 = 20x + 600
–600 + 1200 = 20x –10x
600 = 10x 60 = x
Linear Word-Problems
Ans: 60 gallons
of 10% brine
105. Simple Interest Formula
The interest rate of an investment is usually given in %. Let
r = interest rate (in %)
Linear Word-Problems
106. Simple Interest Formula
The interest rate of an investment is usually given in %. Let
r = interest rate (in %)
P = principal
Linear Word-Problems
107. Simple Interest Formula
The interest rate of an investment is usually given in %. Let
r = interest rate (in %)
P = principal
I = amount of interest for one year
Linear Word-Problems
108. Simple Interest Formula
The interest rate of an investment is usually given in %. Let
r = interest rate (in %)
P = principal
I = amount of interest for one year
then r*P = I.
Linear Word-Problems
109. Simple Interest Formula
The interest rate of an investment is usually given in %. Let
r = interest rate (in %)
P = principal
I = amount of interest for one year
then r*P = I.
Linear Word-Problems
Example I. We have $3000 in a saving account that gives 6%
annual interest. How much interest will be there after one year?
110. Simple Interest Formula
The interest rate of an investment is usually given in %. Let
r = interest rate (in %)
P = principal
I = amount of interest for one year
then r*P = I.
Linear Word-Problems
Example I. We have $3000 in a saving account that gives 6%
annual interest. How much interest will be there after one year?
r = interest rate =
100
6
111. Simple Interest Formula
The interest rate of an investment is usually given in %. Let
r = interest rate (in %)
P = principal
I = amount of interest for one year
then r*P = I.
Linear Word-Problems
Example I. We have $3000 in a saving account that gives 6%
annual interest. How much interest will be there after one year?
r = interest rate =
P = principal
100
6
112. Simple Interest Formula
The interest rate of an investment is usually given in %. Let
r = interest rate (in %)
P = principal
I = amount of interest for one year
then r*P = I.
Linear Word-Problems
Example I. We have $3000 in a saving account that gives 6%
annual interest. How much interest will be there after one year?
r = interest rate =
P = principal
I = amount of interest
100
6
113. Simple Interest Formula
The interest rate of an investment is usually given in %. Let
r = interest rate (in %)
P = principal
I = amount of interest for one year
then r*P = I.
100
6
* 3000
Linear Word-Problems
Example I. We have $3000 in a saving account that gives 6%
annual interest. How much interest will be there after one year?
r = interest rate =
P = principal
I = amount of interest
100
6
Hence r*P =
114. Simple Interest Formula
The interest rate of an investment is usually given in %. Let
r = interest rate (in %)
P = principal
I = amount of interest for one year
then r*P = I.
100
6
* 3000
30
Linear Word-Problems
Example I. We have $3000 in a saving account that gives 6%
annual interest. How much interest will be there after one year?
r = interest rate =
P = principal
I = amount of interest
100
6
Hence r*P =
115. Simple Interest Formula
The interest rate of an investment is usually given in %. Let
r = interest rate (in %)
P = principal
I = amount of interest for one year
then r*P = I.
100
6
* 3000 = 6(30) = $180 = interest.
30
Linear Word-Problems
Example I. We have $3000 in a saving account that gives 6%
annual interest. How much interest will be there after one year?
r = interest rate =
P = principal
I = amount of interest
100
6
Hence r*P =
116. Simple Interest Formula
The interest rate of an investment is usually given in %. Let
r = interest rate (in %)
P = principal
I = amount of interest for one year
then r*P = I.
100
6
* 3000 = 6(30) = $180 = interest.
30
Linear Word-Problems
Example I. We have $3000 in a saving account that gives 6%
annual interest. How much interest will be there after one year?
r = interest rate =
P = principal
I = amount of interest
100
6
Hence r*P =
Remark: The general formula for simple interest is P*r*t = I
where t is the number of years.
117. Simple Interest Formula
The interest rate of an investment is usually given in %. Let
r = interest rate (in %)
P = principal
I = amount of interest for one year
then r*P = I.
100
6
* 3000 = 6(30) = $180 = interest.
30
Linear Word-Problems
Example I. We have $3000 in a saving account that gives 6%
annual interest. How much interest will be there after one year?
r = interest rate =
P = principal
I = amount of interest
100
6
Hence r*P =
Remark: The general formula for simple interest is P*r*t = I
where t is the number of years. Note the similarity to the
concentration formula.
118. Example J. (Mixed investments) We have $2000 more in a 6%
account than in a 5% account. In one year, their combined
interest is $560. How much is in each account?
Linear Word-Problems
119. Example J. (Mixed investments) We have $2000 more in a 6%
account than in a 5% account. In one year, their combined
interest is $560. How much is in each account?
We make a table.
Linear Word-Problems
rate principal interest
120. Example J. (Mixed investments) We have $2000 more in a 6%
account than in a 5% account. In one year, their combined
interest is $560. How much is in each account?
We make a table.
Linear Word-Problems
rate principal interest
6%
5%
121. Example J. (Mixed investments) We have $2000 more in a 6%
account than in a 5% account. In one year, their combined
interest is $560. How much is in each account?
We make a table. Let x = amount in the 5% account.
Linear Word-Problems
rate principal interest
6%
5% x
122. Example J. (Mixed investments) We have $2000 more in a 6%
account than in a 5% account. In one year, their combined
interest is $560. How much is in each account?
We make a table. Let x = amount in the 5% account.
Linear Word-Problems
rate principal interest
6% (x + 2000)
5% x
123. Example J. (Mixed investments) We have $2000 more in a 6%
account than in a 5% account. In one year, their combined
interest is $560. How much is in each account?
We make a table. Let x = amount in the 5% account.
Linear Word-Problems
rate principal interest
6% (x + 2000)
5% x
6
100(x+2000)
5
100 x
124. Example J. (Mixed investments) We have $2000 more in a 6%
account than in a 5% account. In one year, their combined
interest is $560. How much is in each account?
We make a table. Let x = amount in the 5% account.
Linear Word-Problems
rate principal interest
6% (x + 2000)
5% x
6
100(x+2000)
5
100 x
Total amount
of interest is
$560
}
125. Example J. (Mixed investments) We have $2000 more in a 6%
account than in a 5% account. In one year, their combined
interest is $560. How much is in each account?
We make a table. Let x = amount in the 5% account.
Linear Word-Problems
rate principal interest
6% (x + 2000)
5% x
6
100(x+2000)
5
100 x
Total amount
of interest is
$560
Hence:
6
100(x + 2000) + 5
100x = 560
}
126. Example J. (Mixed investments) We have $2000 more in a 6%
account than in a 5% account. In one year, their combined
interest is $560. How much is in each account?
We make a table. Let x = amount in the 5% account.
Linear Word-Problems
rate principal interest
6% (x + 2000)
5% x
6
100(x+2000)
5
100 x
Total amount
of interest is
$560
Hence:
6
100(x + 2000) + 5
100x = 560
[ ] * 1006
100(x + 2000) +
5
100x = 560
}
127. Example J. (Mixed investments) We have $2000 more in a 6%
account than in a 5% account. In one year, their combined
interest is $560. How much is in each account?
We make a table. Let x = amount in the 5% account.
Linear Word-Problems
rate principal interest
6% (x + 2000)
5% x
6
100(x+2000)
5
100 x
Total amount
of interest is
$560
Hence:
6
100(x + 2000) + 5
100x = 560
[ ] * 1006
100(x + 2000) +
5
100x = 560
10011
}
128. Example J. (Mixed investments) We have $2000 more in a 6%
account than in a 5% account. In one year, their combined
interest is $560. How much is in each account?
We make a table. Let x = amount in the 5% account.
Linear Word-Problems
rate principal interest
6% (x + 2000)
5% x
6
100(x+2000)
5
100 x
Total amount
of interest is
$560
Hence:
6
100(x + 2000) + 5
100x = 560
[ ] * 1006
100(x + 2000) +
5
100x = 560
10011
6(x + 2000) + 5x = 56000
}
129. Example J. (Mixed investments) We have $2000 more in a 6%
account than in a 5% account. In one year, their combined
interest is $560. How much is in each account?
We make a table. Let x = amount in the 5% account.
Linear Word-Problems
rate principal interest
6% (x + 2000)
5% x
6
100(x+2000)
5
100 x
Total amount
of interest is
$560
Hence:
6
100(x + 2000) + 5
100x = 560
[ ] * 1006
100(x + 2000) +
5
100x = 560
10011
6(x + 2000) + 5x = 56000
6x + 12000 + 5x = 56000
}
130. Example J. (Mixed investments) We have $2000 more in a 6%
account than in a 5% account. In one year, their combined
interest is $560. How much is in each account?
We make a table. Let x = amount in the 5% account.
Linear Word-Problems
rate principal interest
6% (x + 2000)
5% x
6
100(x+2000)
5
100 x
Total amount
of interest is
$560
Hence:
6
100(x + 2000) + 5
100x = 560
[ ] * 1006
100(x + 2000) +
5
100x = 560
10011
6(x + 2000) + 5x = 56000
6x + 12000 + 5x = 56000
11x = 56000 – 12000
}
131. Example J. (Mixed investments) We have $2000 more in a 6%
account than in a 5% account. In one year, their combined
interest is $560. How much is in each account?
We make a table. Let x = amount in the 5% account.
Linear Word-Problems
rate principal interest
6% (x + 2000)
5% x
6
100(x+2000)
5
100 x
Total amount
of interest is
$560
Hence:
6
100(x + 2000) + 5
100x = 560
[ ] * 1006
100(x + 2000) +
5
100x = 560
10011
6(x + 2000) + 5x = 56000
6x + 12000 + 5x = 56000
11x = 56000 – 12000 = 44000
}
132. Example J. (Mixed investments) We have $2000 more in a 6%
account than in a 5% account. In one year, their combined
interest is $560. How much is in each account?
We make a table. Let x = amount in the 5% account.
Linear Word-Problems
rate principal interest
6% (x + 2000)
5% x
6
100(x+2000)
5
100 x
Total amount
of interest is
$560
Hence:
6
100(x + 2000) + 5
100x = 560
[ ] * 1006
100(x + 2000) +
5
100x = 560
10011
6(x + 2000) + 5x = 56000
6x + 12000 + 5x = 56000
11x = 56000 – 12000 = 44000
x = 44000/11
}
133. Example J. (Mixed investments) We have $2000 more in a 6%
account than in a 5% account. In one year, their combined
interest is $560. How much is in each account?
We make a table. Let x = amount in the 5% account.
Linear Word-Problems
rate principal interest
6% (x + 2000)
5% x
6
100(x+2000)
5
100 x
Total amount
of interest is
$560
Hence:
6
100(x + 2000) + 5
100x = 560
[ ] * 1006
100(x + 2000) +
5
100x = 560
10011
6(x + 2000) + 5x = 56000
6x + 12000 + 5x = 56000
11x = 56000 – 12000 = 44000
x = 44000/11 = $4000
}
134. Example J. (Mixed investments) We have $2000 more in a 6%
account than in a 5% account. In one year, their combined
interest is $560. How much is in each account?
We make a table. Let x = amount in the 5% account.
Linear Word-Problems
rate principal interest
6% (x + 2000)
5% x
6
100(x+2000)
5
100 x
Total amount
of interest is
$560
Hence:
6
100(x + 2000) + 5
100x = 560
[ ] * 1006
100(x + 2000) +
5
100x = 560
10011
6(x + 2000) + 5x = 56000
6x + 12000 + 5x = 56000
11x = 56000 – 12000 = 44000
x = 44000/11 = $4000
}
So we've $4000 in the 5% accnt.
135. Example J. (Mixed investments) We have $2000 more in a 6%
account than in a 5% account. In one year, their combined
interest is $560. How much is in each account?
We make a table. Let x = amount in the 5% account.
Linear Word-Problems
rate principal interest
6% (x + 2000)
5% x
6
100(x+2000)
5
100 x
Total amount
of interest is
$560
Hence:
6
100(x + 2000) + 5
100x = 560
[ ] * 1006
100(x + 2000) +
5
100x = 560
10011
6(x + 2000) + 5x = 56000
6x + 12000 + 5x = 56000
11x = 56000 – 12000 = 44000
x = 44000/11 = $4000
}
So we've $4000 in the 5% accnt. and $6000 in the 6% accnt.
136. Exercise A. Express each expression in terms of the given x.
Linear Word-Problems
Let x = $ that Joe has.
1. Mary has $10 less than what Joe has, how much does Mary have?
2. Mary has $10 more than twice what Joe has, how much does Mary have?
3. Mary has twice the amount that’s $10 more than what Joe has, how much does
Mary have?
4. Together Mary and Joe have $100, how much does Mary have?
5. Mary has $100 less than what Joe has, what is twice what Joe and Mary have
together? Simplify your answer.
6. Mary has $100 more than Joe, what is half of what Joe and Mary have together?
Simplify your answer.
B. Peanuts cost $3/lb, cashews cost $8/lb. We have x lbs of peanuts.
Translate each quantity described below in terms of x.
7. What is the cost for the peanuts?
8. We have 4 more lbs of cashews as peanuts, what is the cost for the cashews?
9. Refer to problem 7 and 8, what is the total cost for all the nuts?
Simplify answer.
10. We have twice as many of cashews as peanuts in weight, what is the total cost
of the nuts? Simplify your answer.
11. We have a total of 50 lb of peanuts and cashews, how many lbs of cashews do
we have? What is the total cost for all the nuts? Simplify you answers.
137. C. Money Sharing Problems. Select the x. Set up an equation then solve.
1. A and B are to share $120. A gets $30 more than B, how much does each get?
2. A and B are to share $120. A gets $30 more than twice what B gets, how much
does each get?
3. A and B are to share $120. A gets $16 less than three times of what B gets, how
much does each get?
4. A, B and C are to share $120. A gets $40 more than what B gets, C gets $10 less
than what B gets, how much does each get?
5. A, B and C are to share $120. A gets $20 more than what twice of B gets, C gets
$10 more than what A gets, how much does each get?
6. A and B are to share $120. A gets 2/3 of what B gets, how much does each get?
7. A, B and C are to share $120. A gets 1/3 of what B gets, C gets ¾ of what B
gets, how much does each get?
D. Wet Mixtures Problems (Make a table for each problem 8 – 19)
8. How many gallons of 20% brine must be added to 30 gallons of 40% brine to
dilute the mixture to a 25% brine?
9. How many gallons of pure water must be added to 40 gallons of 30% brine to
dilute the mixture to a 20% brine?
10. How many gallons of 10% brine must be added with 40% brine to make 30
gallons of 20% brine?
11. How many gallons of 50% brine must be added with pure water to make 30
gallons of 20% brine?
138. 12. We have $3000 more saved in an account that gives 5% interest rate than in an
account that gives 4% interest rate. In one year, their combined interest is $600.
How much is in each account?
13. We have saved at a 6% account $2000 less than twice than in a 4% account. In
one year, their combined interest is $1680. How much is in each account?
14. The combined saving in two accounts is $12,000. One account gives 5%
interest rate, the other gives 4% interest rate. The combined interest is %570 in one
year. How much is in each account?
15. The combined saving in two accounts is $15,000. One account gives 6%
interest rate, the other gives 3% interest rate. The combined interest is $750 in one
year. How much is in each account?
Linear Word-Problems
F. Dry Mixture Problem – These are similar to the wet–mixture problems.
16. Peanuts cost $3/lb, cashews cost $6/lb. How many lbs of peanuts are needed
to mix with12 lbs cashews to make a peanut–cashews–mixture that costs $4/lb?
17. Peanuts cost $3/lb, cashews cost $6/lb. How many lbs of each are needed to
make 12 lbs of peanut–cashews–mixture that costs $4/lb?
18. Peanuts cost $2/lb, cashews cost $8/lb. How many lbs of cashews are needed
to mix with 15 lbs of peanuts to get a peanut–cashews–mixture that costs $6/lb?
19. Peanuts cost $2/lb, cashews cost $8/lb. How many lbs of each are needed to
make 50 lbs of peanut–cashews–mixture that cost $3/lb?
E. Mixed Investment Problems.