# Sample questions

#1. Blocks

Tom is learning the addition and subtraction using small blocks as shown in the figure below. Can you help to find how many blocks are left after subtraction?

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Explanation: Blocks Back to the problem

Initially there is one hundred three tens and eight ones. Out of these, two tens and six ones are taken out. One hundred, one ten, and two ones are left; which are equal to 112.

#2. Pizza slices

Riya’s mom had brought one cheese pizza. She cut it into four equal slices. She gave one-fourth to Riya and another one-fourth to her brother. How much pizza was left?

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Explanation: Pizza slices Back to the problem

Out of the four slices of the whole pizza, two slices are given to Riya and her brother. The balance pizza is two-fourths or one-half of the whole.

#3. House

Peter made a house using arrows and circles. How many more arrows than circles were used in this house?

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Explanation: House Back to the problem

Number of arrows = 14. Number of circles = 12. There are two arrows more than circles.

#1. Summer camp

A group of students went to a summer camp. The group left their camp on May 27th at 8:15 am and returned to the camp on 7th June at 8:30 pm. How many nights were they away from the camp?

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Explanation: Summer camp Back to the problem

The group was back on 7th June; so that day is excluded from the counting. The month of May has 31 days and so include 31st May also counting.

#2. Equivalent Fraction

Which of the following fractions is equivalent to \frac13?

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Explanation: Equivalent Fraction Back to the problem

You can generate equivalent fractions by multiplying the numerator and the denominator by the same number.

\frac13,\frac26,\frac39,\frac4{12},\frac5{15},\frac6{18}... are equivalent fractions.

\frac13\times\frac55=\frac5{15}

#3. Triangles

How many triangles of all sizes are there in the picture below?

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Explanation: Triangles Back to the problem

Four triangles are clearly visible at the four corners of the outer square. The inner square is divided into two bigger triangles by the horizontal line, and similarly by the vertical line. The inner square is also divided into four triangles by the two lines.

4 + 2 + 2 + 4 = 12

#1. Decimal number

Which of the following decimal numbers lies between the fractions \frac35 and \frac45?

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Explanation: Decimal number Back to the problem

\frac35=\frac6{10}=0.60 and \frac45=\frac8{10}=0.80.

The decimal number 0.67 is within the interval (0.60, 0.80). Thus it lies somewhere between the fractions \frac35 and \frac45.

#2. Sand

At the sea beach, James and Mike were playing with sand. James made a square with each side of 16 inches. Mike made a rectangle with a length of 32 inches. If the areas of the figures made by them were equal, what was the width of Mike’s rectangle?

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Explanation: Sand Back to the problem

Area of the square = 16 x 16 = 256 sq. inches.
As area of the square and the rectangle are equal, so the area of the rectangle is also 256 sq. in.
The area of the rectangle = length x width
Width = area/ length = 256/32 = 8 inches

#3. Symmetry

Which of the following has only one axis of symmetry?

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Explanation: Symmetry Back to the problem

The Euro image has only one line of symmetry.

A triangle has three and a square has four lines of symmetry. The Pond and the Dollar signs do not have any lines of symmetry.

#1. ABC

A triangle ABC has been plotted on a coordinate graph as above. How many of the following points will not be within the area of the triangle?

(2, 2); (2, 3); (3, 3); (4, 1); (4, 2); (5, 1); (5, 2)

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Explanation: ABC Back to the problem

The points (2, 2),(4, 2), and (5, 2) fall within the triangle. But (2, 3), (3, 3), (4, 1), and (5, 1) are outside the boundary of the triangle.

#2. A water tank

A water tank has marble cubes flooring as shown in the picture below. Each side of the marble cube measures one foot.

How much water will be required to fill in the tank?

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Explanation: A water tank Back to the problem

First think of the tank before the flooring was done and workout its measurements. Each side of the cube is one feet. The front view shows a length will be 6 ft and a width 3 ft. Now look at the marker on the right side of the tank. This indicates the total depth was 4 ft, out of which 1 ft has been used for flooring. So net depth for water is 3 ft only.

Net Capacity  of tank available for holding water = 6 ft x 3 ft x 3 ft = 54 cubic feet

#3. Digits

Below is an example of displaying an addition written in codes. Each of the Greek letters α, β, γ and δ represents one digit. Which digits are represented by the codes (α, β, γ, δ)?

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Explanation: Digits Back to the problem

It is given the total is a 4-digit number with all digits being the same. So the possible options may be 1111, 2222, 3333, 4444……..9999. Further, the total is the sum of a 3-digit number and two equal numbers, each of two digits.

Examine possibilities. For example, sum of the largest 2-digit numbers is 99 + 99= 199. If you take 888 as a 3-digit number, the total comes to 199 + 888 = 1087. All digits are not the same.

But the lowest 4-digit (with all digits being the same) is 1111.  So the possible 3-digit number (with all digits being the same) needs to be higher than 888. Thus it is 999.

1111 - 999 = 112. It gives the sum of both 2-digit numbers, 56 each.

#1. Alloy

An alloy is made up of multiple metals each with different ratios. An engineer developed the expression 60+3(a^2-b)-12b for calculating the weight of the alloy based on the weights of two metals a and b. If a = 24 grams and b = 28 grams, what is the weight of the alloy?

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Explanation: Alloy Back to the problem

Replace a = 24 and b = 28 grams.

60+3(a^2-b)-12b=60+3(24^2-28)-12(28)
=60+3(576-28)-336
=60+1644-336
=1368 grams

#2. Prism

The figure below displays the net of a rectangular prism, where the dimensions of the rectangles are 8 in. and 20 in. What is the total surface area of the prism?

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Explanation: Prism Back to the problem

Surface area of the prism made up of 2 squares of 8 in. by 8 in., and four rectangles of 8 in. by 20 in.

Area of squares = 2(8)(8) = 128 in.^2
Area of rectangles = 4(8)(20) = 640 in.^2
Total surface area 1= 28 + 640 = 768 in.^2

#3. XY

In the following operation, XX and YX are two different 2-digit numbers. What are XX and YX?

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Explanation: XY Back to the problem

X is either 0 or 5. Digit 0 cannot be used. So, only option is X = 5. Replace X = 5.

So, Y must be 4.

#1. Hanging weights

Hanging different weights from exactly the same springs increases the lengths of the springs as shown below:

The magnitudes of increases in the length of the springs based on different weights are shown in the table below:

What is the ratio of the increase in length with respect to the increase in weight?

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Explanation: Hanging weights Back to the problem

Ratio of the first pair =\frac25

Ratio of the second pair =\frac4{10} = \frac25

Ratio of the third pair =\frac6{15} = \frac25

Ratio of the fourth pair =\frac8{20} = \frac25

Ratio of the fifth pair =\frac{10}{25} = \frac25

Ratio of the sixth pair =\frac{12}{30} =\frac25

Comparison of the ratios results in
\frac25=\frac4{10}=\frac6{15}=\frac8{20}=\frac{10}{25}=\frac{12}{30}

Therefore, the ratio is \frac25

#2. Equation

The following equation is developed by an engineer to find the loss of pressure of a liquid flowing through a pipe at a certain point. What is the amount of a loss of pressure?

0.22P-\frac34(24P-12)=252.3-34P

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Explanation: Equation Back to the problem

0.22P-\frac34(24P-12)=252.3-34P
0.22P-18P+9=252.3-34P
0.22P-18P+34P=252.3-9
16.22P=243.3
P=15

#3. Triangle

In the following right triangle, the numbers on three sides of the triangle are related by the same rule. What is the missing number?

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Explanation: Triangle Back to the problem

Square of one number from the hypotenuse is equal to the sum of the squares of two numbers on the legs. These triples are listed below:

3 – 4 – 5
6 – 8 – 10
9 – 12 – 15

Therefore, the sum of squares of 5 and 12 must be the square of number on the hypotenuse.
This number is 13.

#1. Expression

Which expression is equivalent to (2^3)\div(5^2-3^2)-4^3(3^3-5^2) ?

I. (2^3)\div(6)-4^3(8)

II. (2^3)\div(5^2-3^3)-(4^3)(3^3)-5^2

III. (2^3)\div(5^2)-3^2-4^3(3^3-5^2)

IV. 2^{3-4}-2^{6+1}

V. 2^{4-3}-2^{3-1}

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Explanation: Expression Back to the problem

(2^3)\div(5^2-3^2)-4^3(3^3-5^2)
=(2^3)\div(25-9)-4^3(27-25)
=\left(2^3\right)\div\left(2^4\right)-2^6(2)
=2^{3-4}-2^{6+1}

#2. Angle F

In the following figure, AB and CD are parallels, and the measures of two angles are given in the figure. Which is the measure of the angle F?

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Explanation: Angle F Back to the problem

Draw FH parallel to the given parallel lines.

a and 32 are the measures of two alternate interior angles. So, they are equal.
(1)  a = 32

b and 58 are the measures of two alternate interior angles. So, they are equal.
(2)  b = 58

a + b = 32 + 58
F = 90 degrees

#3. Pattern

The following operations on numbers follow a general algebraic pattern:

21\times19=(20+1)(20-1)=20^2-1^2=399
201\times199=(200+1)(200-1)=200^2-1^2=39999
3001\times2999=(3000+1)(3000-1)=3000^2-1^2=8,999,999

Which is the equivalent of 5002\times4998?

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Explanation: Pattern Back to the problem

5002\times4998=(5000+2)(5000-2)=5000^2-4