Section A
Multiple Choice Questions — 15 Questions
(+4 / -1)
1
arithmetic
easy
Solve: 826 − (626 − 529) + 201
A
900
B
725
C
1127
D
930
E
1124
Show answer
Answer: D
Regrouping strategy:
(826 − 626) + (529 + 201)
= 200 + 730 = 930.
2
place-value
medium
What is the digit in the hundreds place of the sum below?
6969 + 9696 + 6996 + 9669
A
6
B
9
C
0
D
2
E
3
Show answer
Answer: E
For each place, we have digits 6 + 6 + 9 + 9 = 30.
Sum = 30000 + 3000 + 300 + 30 = 33330.
So, the hundreds digit is 3.
3
number-theory
medium
What is the smallest four-digit number that is divisible by 7 and 5?
A
1007
B
1010
C
1035
D
1015
E
1050
Show answer
Answer: D
A number that is divisible by 7 and 5 must be divisible by 35.
The smallest 4-digit number is 1000.
1000 ÷ 35 = 28 R 20.
Thus, 29 × 35 = 1015 is the smallest four-digit number divisible by 5 and 7.
4
number-theory
hard
The product of all the digits of a 3-digit number is 12. How many such 3-digit numbers are there?
A
15
B
12
C
9
D
11
E
18
Show answer
Answer: A
12 = 1×2×6 = 1×3×4 = 2×2×3.
Numbers formed by (1,2,6) - 126, 162, 216, 261, 612, 621 — 6 numbers.
Numbers formed by (1,3,4) - 134, 143, 314, 341, 413, 431 — 6 numbers.
Numbers formed by (2,2,3) - 223, 232, 322 — 3 numbers.
So, there are 6 + 6 + 3 = 15 such numbers.
5
algebra
medium
Josh thought of a two-digit number and then added a zero at the end of that number to make it a three-digit number. He then added the three-digit number to his original number and the result was 726. What was his two-digit number?
A
76
B
67
C
81
D
66
E
58
Show answer
Answer: D
Perform addition:
A B 0
+ A B
-------
7 2 6
It is obvious that B = 6. Then B + A = 6 + A = 12 or A = 6.
Therefore, the original two-digit number is 726 ÷ 11 = 66.
6
patterns
medium
If one stack of coins has 8 coins and they are arranged in such a way that the first layer has one stack, the second layer has 2 stacks, and so on. Find how many coins are required to make a total of 5 layers of coin stacks.
A
120
B
150
C
210
D
48
E
75
Show answer
Answer: A
The number of coin stacks in 5 layers = 1 + 2 + 3 + 4 + 5 = 15.
Total number of coins required to make 5 layers = 15 × 8 = 120.
7
calendar
medium
Sara's birthday falls on Friday, 15th August this year. Which day will her birthday fall on next year if next year is a leap year?
A
Tuesday
B
Thursday
C
Sunday
D
Monday
E
Saturday
Show answer
Answer: C
One year has 52 weeks and 1 extra day. If next year is a leap year, there will be 52 weeks and 2 extra days.
So, if 15th August this year is Friday, 15th August next year, which is a leap year, will fall on Sunday, which is 2 days after Friday.
8
money
medium
An eraser costs $5 less than a pencil. If 5 pencils and 5 erasers cost $40, what is the price of 2 pencils?
A
$6.50
B
$1.50
C
$130
D
$13
E
$65
Show answer
Answer: D
5 pencils and 5 erasers = $40.
1 pencil and 1 eraser = $40 ÷ 5 = $8.
1 eraser = 1 pencil − $5, or 1 pencil = 1 eraser + $5.
1 eraser + $5 + 1 eraser = $8.
1 eraser = ($8 − $5) ÷ 2 = $1.50.
1 pencil = $1.50 + $5 = $6.50.
2 pencils = $6.50 × 2 = $13.
9
algebra
medium
If the product of two numbers is 154 and their sum is 25, find the smaller of the 2 numbers.
A
7
B
22
C
14
D
11
E
4
Show answer
Answer: D
154 = 2 × 7 × 11.
All possible solutions to get 154 as a product of 2 numbers:
2 × 77, sum 79; 7 × 22, sum 29; 11 × 14, sum 25.
So, the two numbers that give a product of 154 and a sum of 25 are 11 and 14.
The smaller number is 11.
10
arithmetic
medium
Henry was driving a car in such a way that after every 10 minutes he increased his speed and covered 1 km extra than the previous 10 minutes. If he drove 2 km in the first 10 minutes, how many km did he cover in an hour?
A
27
B
20
C
12
D
24
E
25
Show answer
Answer: A
An hour has 60 minutes. 60 ÷ 10 = 6 groups of 10 minutes.
First 10 minutes he drove 2 km.
Second 10 minutes he drove 2 + 1 = 3 km, and so on.
Thus, he drove a total of 2 + 3 + 4 + 5 + 6 + 7 = 9 × 3 = 27 km in an hour.
11
combinatorics
medium
If three students were to pick one ball each from a bag with 3 different coloured balls, how many different ways are there?
A
6
B
3
C
8
D
9
E
4
Show answer
Answer: A
Consider students 1, 2 and 3 and colours R, G and B. Listing all the ways each student can be assigned a different colour: (R,B,G), (R,G,B), (G,R,B), (G,B,R), (B,R,G), (B,G,R).
As shown, there are 6 different ways in total.
12
patterns
medium
What is the 11th term in the pattern below? 17, 20, 25, 32, 41, ...
A
116
B
137
C
126
D
160
Show answer
Answer: B
Notice that the pattern is +3, +5, +7, +9, +11, ...
Thus, the sequence is: 17, 20, 25, 32, 41, 52, 65, 80, 97, 116, 137.
13
place-value
medium
A new number is formed by placing a zero between 3 and 7 of the number 23791. Find the difference between the two numbers.
A
2070
B
20700
C
207000
D
20710
E
20791
Show answer
Answer: C
The difference of 230791 − 23791 is the same as the difference of 230000 − 23000 = 207000.
14
geometry
medium
A rectangle is formed by placing two squares of the same size next to each other. If the perimeter of the square is 60 cm, find the area of the rectangle in cm².
A
288
B
450
C
354
D
360
E
225
Show answer
Answer: B
Length of the square = perimeter ÷ 4 = 60 ÷ 4 = 15 cm.
Length of rectangle = 15 + 15 = 30 cm.
Breadth of rectangle = 15 cm.
Thus, the area of the rectangle = 30 × 15 = 450 cm².
15
patterns
hard
Radha has to cut small strips of ribbon of 5 cm each. But after every 3 strips of 5 cm, she cuts a strip of 4 cm mistakenly. If a total roll of 100 cm was cut into small ribbons, how many 5 cm ribbon strips are there?
A
15
B
16
C
14
D
19
E
20
Show answer
Answer: B
Notice the pattern of strips being cut into: 5 cm, 5 cm, 5 cm, 4 cm, 5 cm, 5 cm, ...
So, one set can be considered as 5 + 5 + 5 + 4 = 19 cm long.
100 ÷ 19 = 5 R 5 → 5 sets of 19 cm and one 5 cm.
Thus, there are 5 × 3 + 1 = 16 strips which are 5 cm long.
Section B
Open-Ended (Integer) Questions — 10 Questions
(+5)
16
geometry
medium
A 4 cm × 4 cm × 4 cm cube is formed by arranging small cubes with dimensions 1 cm × 1 cm × 1 cm. If the big cube is painted blue on all faces, how many small cubes have only 1 side painted?
Show answer
Answer: 24
Consider one face as a 4 cm × 4 cm square. There are 16 small squares. We notice that only the middle 4 small cubes have only one side visible and thus have only one side painted.
There are 6 faces in total in a cube, so there are 6 × 4 = 24 small cubes with one side painted.
17
ratio
medium
Each student ate three-fifth of a pizza. A total of 18 pizzas were ordered. How many students were there?
Show answer
Answer: 30
3/5 of a pizza = 1 student.
3 pizzas = 5 students.
3 × 6 = 18 pizzas = 5 × 6 = 30 students.
18
fractions
medium
One-third of Tom's total score was scored in English, two-fourth was scored in Maths and the remaining he scored 45 in French. What was his total score?
Show answer
Answer: 270
Two-fourth is the same as half. 1/3 + 1/2 = 5/6, and the remaining one-sixth of the total score is 45.
Thus, the total score = 45 × 6 = 270.
19
algebra
hard
The combined age of David and Harry is 50 years old now. If 10 years ago, David was twice as old as Harry, what will be Harry's age 5 years from now?
Show answer
Answer: 25
Combined age now = 50.
The combined age 10 years ago was 50 − 10 − 10 = 30.
Since David was twice as old as Harry 10 years ago, let Harry's age 10 years ago be 1 unit and David's age 10 years ago be 2 units.
1 unit + 2 units = 3 units = 30.
Harry was 30 ÷ 3 = 10 years old 10 years ago.
Harry's age 5 years from now = 10 + 10 + 5 = 25 years old.
20
number-theory
hard
If 300 digits were used to print the page numbers of a book, how many pages does the book have?
Show answer
Answer: 136
The number of digits used to print 1-digit page numbers (page 1 to 9) = 9.
The number of digits used to print 2-digit page numbers (page 10 to 99) = 90 × 2 = 180.
The number of digits used to print the first 99 pages = 180 + 9 = 189.
300 − 189 = 111 digits were used to print the remaining 3-digit page numbers.
111 ÷ 3 = 37 3-digit page numbers starting from page 100.
Thus, the book has 99 + 37 = 136 pages.
21
number-theory
hard
How many numbers will have the same quotient and remainder when divided by 12?
Show answer
Answer: 12
Notice that any number when divided by 12 can have remainders of 0, 1, 2, 3, ..., 11 but not 12. The numbers that will have the same quotient and remainder when divided by 12 are:
12×0+0=0, 12×1+1=13, 12×2+2=26, 12×3+3=39, 12×4+4=52, ...
The last possible number is 12×11+11=143, as 11 is the largest possible remainder when divided by 12.
Thus, there are only 12 numbers which will give the same quotient and same remainder when divided by 12.
22
patterns
hard
What will be the sum of the first 50 numbers of the following sequence? 7, 13, 27, 33, 7, 13, 27, 33, ...
Show answer
Answer: 980
Notice that the 4 same numbers are repeating themselves in the pattern.
Sum of 1 group of 4 numbers: 7 + 13 + 27 + 33 = 80.
Also, 50 ÷ 4 = 12 R 2.
So, there are 12 groups of the 4 numbers with the sum of 80, followed by the first 2 numbers in the group, i.e. 7 and 13.
Thus, sum of first 50 numbers = 80 × 12 + 7 + 13 = 960 + 20 = 980.
23
logic-puzzle
hard
There are three boxes labelled with numbers 1, 2 and 3. Each box contains different numbers of different coloured marbles.
• The number of yellow marbles is half of the sum of red and blue marbles.
• There are more blue marbles than red marbles.
• Box 1 has 10 marbles.
• The total number of marbles is 60.
How many blue marbles are there?
Show answer
Answer: 30
Since there are a total of 60 marbles and the number of yellow marbles is half of the sum of red and blue marbles, the number of yellow marbles = 20, while the total number of red and blue marbles = 40.
Also, given that red marbles are less than blue marbles and Box 1 has 10 marbles, Box 1 must have 10 red marbles.
Thus, the remaining 30 marbles are blue.
24
number-theory
medium
Given that k × 726 is a square of 66, find the value of k.
Show answer
Answer: 6
k × 726 = 66 × 66 = 6 × 11 × 6 × 11.
726 = 6 × 11 × 11.
Thus, k must be 6.
25
number-theory
medium
A box of apples can be shared equally between 3, 4, 5 or 6 classes. What is the smallest possible number of apples in the box if it had more than 200 but less than 500 apples?
Show answer
Answer: 240
The smallest number divisible by 3, 4, 5 and 6 is 60.
Multiples of 60 are 60, 120, 180, 240, 300, ...
So, the minimum number of apples in the box which has more than 200 apples is 240.