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- Compare Quantity A and Quantity B and select one of the following four answer choices:

2^{15}

3^{10}

- Quantity A is greater
- Quantity B is greater
- The two quantities are equal
- The relationship cannot be determined from the information given

Answer: B

In order to compare the two quantities A and B, we need to make either the bases the same (so that we can compare the indices) or make the indices (exponents) the same (and compare the bases). 2 and 3 are prime numbers so the bases will never be the same, let’s equate the indices and then compare the bases to see which is larger.

2^{15} = (2^{3})^{5} and 3^{10} = (3^{2})^{5}

QUANTITY A = 8^{5}

QUANTITY B = 9^{5}

Quantity B is greater.

- In 2008, the list price of a home was 1/3 greater than the original price. In 2018, the list price of the home was 1/2 greater than the original price. By what percent did the list price of the home increase from 2008 to 2018?

- 10.00%
- 12.50%
- 16.66%
- 33.33%
- 50.00%

Answer: B

First, let’s look at the standard school method of solving this question.

Let the original price be x

In 2008, the list price of a home was 1/3 greater than the original price = x + x/3 = 4x/3

In 2018, the list price of the home was 1/2 greater than the original price = x + x/2 = 3x/2

Increase = 3x/2 – 4x/3 = x/6

Percent increase from 2008 to 2018 = (x/6 ÷ 4x/3) x 100 = 12.5%.

OR

If working with fractions is cumbersome for you, then here’s another solution.

Let the original price be 300 (or any number that can be easily divided by both 3 and 2)

In 2008, the list price of a home was 1/3 greater than the original price = 300 + (300/3) = 400

In 2018, the list price of the home was 1/2 greater than the original price = 300 + (300/2) = 450

Increase = 450 – 400 = 50

Percent increase from 2008 to 2018 = (50/400) x 100 = 12.5%.

- The figure below represents a square photograph bordered by a frame that has a uniform width of 3 inches. If the frame and the picture have the same area, and each of the photograph’s sides measures x inches, which of the following equations is true?

- (x +6)
^{2}= 4x^{2} - (x – 3)
^{2}= 2x^{2} - (x – 6)
^{2}= 4x^{2} - (x + 3)
^{2}= 2x^{2} - (x + 6)
^{2}= 2x^{2}

Answer: E

Each of the photograph’s sides measures x inches, area of photograph = x^{2.}

Length of outer square (photograph + frame) = x + 3+ 3 = x + 6

Area of photograph + frame = (x+6)^{2}

If the frame and the picture have the same area, then

Area of photograph + frame = 2 x Area of photograph

(x+6)^{2} = 2x^{2}

- On the xy-plane, the center of circle O is at point (4,3). If the point (11,3) lies outside of the circle and the point (4,9) lies inside of the circle, which of the following could be the radius of the circle?

- 5.0
- 5.5
- 6.0
- 6.5
- 7.0

Answer: D

Observe how the points (4.3) and (11,3) are in a straight line and so are (4,3) and (4,9), which means we can save some time and not use the distance formula. A rough sketch would be useful.

OQ = 6 units (9-3=6) and is inside the circle hence smaller than the radius.

OP = 7 units (9-3=6) and is outside the circle hence larger than the radius.

So the radius is more than 6 units but less than 7 units. Only possible answer is 6.5 units.

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- 3a + 5
- 3a – 9
- 3a + 10
- 3a – 4
- 3a + 4

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