1. In the following figure, the black ball moves one position at a time
clockwise. The white ball moves two positions at a time counter-
clockwise, In how many moves will they meet again?
Solution:
Jot down the location of the balls for every movement:
(black, white): (B, F)
(C, D)
(D, B)
(E, G)
(F, E)
(G, C)
(A, A)
Answer: 7 moves
2. There are five positive numbers. The sum of the first and the fifth number
is 13. The second number is one-third of the sum of these five numbers,
the third number is one-fourth of this sum and the fourth number is one-
fifth of this sum. What is the value of the largest number?
Solution:
Let’s call the five numbers as: 𝑎, 𝑏, 𝑐, 𝑑, 𝑒;
and the sum as: 𝐴 = 𝑎 + 𝑏 + 𝑐 + 𝑑 + 𝑒
Given 𝑎 + 𝑒 = 13
𝑏 =
1
3
𝐴
𝑐 =
1
4
𝐴
𝑑 =
1
5
𝐴
𝐴 =
1
3
𝐴 +
1
4
𝐴 +
1
5
𝐴 + 13
=
47
60
𝐴 + 13
13
60
𝐴 = 13
𝐴 = 60
𝑏 = 20, 𝑐 = 15, 𝑑 = 12
Answer: the largest number is 20
3. Solution 1:
Since all four footprints were at the top and the bottom, the number of
steps = 𝐿𝐶𝑀[2,3,4,5] = 60.
And since 4 is a multiple of 2, whichever steps Jessica steps on must have
been stepped on by Peter. So we just need to look at footprints from Peter
(2 steps), Bruce (3 steps) and Maitrevi (5 steps).
Steps with footprints from:
P, B and M =
60
2×3×5
= 2
P and B =
60
2×3
= 10
P and M =
60
2×5
= 6
B and M =
60
3×5
= 4
Four friends were racing side by side down a dusty staircase. Peter went
down two steps at a time, Bruce three steps at a time, Jessica four steps at
a time, and Maitreyi five steps at a time. If the only steps with all four
footprints were at the top and the bottom, how many steps had only one
person’s footprint?
Continue on next page …
4. Solution 1 continue:
Steps with footprints from:
P =
60
2
= 30 (which includes J)
B =
60
3
= 20
M =
60
5
= 12
16 steps with P footprints
Out of these 16 steps,
8 have footprints from J.
Therefore, steps with only one footprints
= 8 + 8 + 4
= 20
5. Solution 2:
Since all four footprints were at the top and the bottom, the number of
steps = 𝐿𝐶𝑀[2,3,4,5] = 60.
Answer: 20 steps with 1 person’s footprints
6. Solution:
𝑀 + 𝑇𝑢 + 𝑊 = 115 −− − ①
𝑊 + 𝑇ℎ = 85 −− − ②
𝑇𝑢 + 𝑇ℎ = 90 −− − ③
𝑀 + 𝐹 = 70 −− −④
𝑇ℎ + 𝐹 = 80 −− −⑤
The following table shows the number of mathematics books sold over a
period of five days. Find the number of books sold on Tuesday.
① - ② 𝑀 + 𝑇𝑢 – 𝑇ℎ = 30 −− − ⑥
④ - ⑤ 𝑀 − 𝑇ℎ = −10 −− − ⑦
Replace ⑦ into ⑥, 𝑇𝑢 – 10 = 30
𝑇𝑢 = ___
Answer: 40
