# PSAT Math : Arithmetic

## Example Questions

### Example Question #61 : Proportion / Ratio / Rate

In 7 years Bill will be twice Amy’s age. Amy was 1.5 times Molly’s age 2 years ago. If Bill is 29 how old is Molly?

8

12

9

6

5

8

Explanation:

Consider

(Bill + 7) = 2 x (Amy + 7)

(Amy – 2) = 1.5 x (Molly – 2)

Solve for Molly using the two equations by finding Amy’s age in terms of Molly’s age.

Amy = 2 + 1.5 Molly – 3 = 1.5 x Molly – 1

Substitute this into the first equation:

(Bill + 7) = 2 x (Amy + 7) = 2 x (1.5 x Molly – 1 + 7) = 2 x (1.5 x Molly + 6) = 3 x Molly + 12

Solve for Molly:

Bill + 7 – 12 = 3 x Molly

Molly = (Bill – 5) ¸ 3

Substitute Bill = 29

Molly = (Bill – 5) ¸ 3 = 8

### Example Question #62 : Proportion / Ratio / Rate

When Christina opens a bag of white and milk chocolate pieces, 20% of the chocolate pieces are white. After Christina eats 10 milk chocolate pieces, the ratio of brown chocolate to white chocolate is 2 to 3. How many pieces of chocolate are left in the bag?

5

15

12

3

2

5

Explanation:

Let original white chocolate pieces = W and original milk chocolate pieces = M. So the total number of pieces in the original bag is M + W.

From the first sentence: (M + W) x 0.2 = W or

0.2 M = 0.8 W or [M = 4W]

Once Christina has eaten 10 milk chocolate pieces, there are W pieces of white chocolate, (M – 10) pieces of milk chocolate and (M + W – 10) pieces total. According to the second sentence:

W ¸ (M – 10) = 3 ¸ 2

Or 2W = 3M - 30

Insert the equation in brackets: 2W = 3[4W] + 30 = 12W – 30

10W = 30 or W = 3 and M = 12

We want “How many pieces of chocolate are left in the bag” or (M – W – 10).

So (M +W – 10) = 3 + 12 – 10 = 5

### Example Question #63 : Proportion / Ratio / Rate

You are buying a new car.  The car gets 33 miles per gallon in the city and 39 miles per gallon on the highway.  You plan on driving 30,000 miles over three years and 10,000 of that will be city driving.  If gas costs $3.50 per gallon, how much will you pay in gas over the three year period (round to the nearest cent)? Possible Answers:$2692.31

$2855.47$1060.60

$1794.87$3181.81

$2855.47 Explanation: Cost = ( Miles driven / Miles per gallon) * 3.50 Total Mileage = City Miles + Highway Miles 30,000 = 10,000 + Highway Miles Highway Miles = 20,000 Cost City = ( 10,000 miles / 33 miles per gallon ) * 3.50 Cost City = 303.03 *$3.50 = $1060.60 Cost Highway = ( 20,000 miles / 39 miles per gallon ) * 3.50 Cost Highway = 512.82 *$3.50 = $1794.87 Total Cost = Cost City + Cost Highway =$1060.60 + $1794.87 =$2855.47

### Example Question #64 : Proportion / Ratio / Rate

A class room of 8th graders is 1/3 boys. Of all the students 4/5 of them are aged 14 while the others are aged 13. If there are 20 girls in the class, approximately how many boys are age 13?

5

6

8

4

2

2

Explanation:

If 20 students are girls, this is 2/3 of the class, giving 30 students total with 10 of them being boys. 4/5 of the boys will be 14, leaving 1/5 of the boys age 13. 1/5 of 10 is 2.

### Example Question #65 : Proportion / Ratio / Rate

x and y are integers such that x > 0 and y > 0 .

12x + 3y = 176,500.

Quantity A: The maximum possible value of x

Quantity B: The maximum possible value of y

The two quantities are equal

Quantity B is greater

The relationship cannot be determined from the information given

Quantity A is greater

Quantity B is greater

Explanation:

Note that it is not necessary to find the solution to this problem. Thus, find an expression for x and an expression for y at their maximums and compare.

First, observe from the equation that x gets smaller as y gets bigger and y gets smaller as y gets bigger. This can be found by making the equation in the form (y = mx + b) and finding that m is negative. Thus there is a negative correlation between x and y.

Therefore at its maximum, x is such that y = 0. In other words,

x = 176,500 / 12

y is maximum when x = 0. In other words,

y = 176,500 / 3

Note that 176,500 / 3 > 176,500 / 12 because the denominator is smaller.

### Example Question #66 : Proportion / Ratio / Rate

If 1015 meters = 1 petameter and 1018 meters = 1 exameter, how many petameters are equal to 1 exameter?

0.1
10000
100
0.01
1000
Explanation:

The problem gives us two conversion ratios, which are (1018 meters/ 1 exameter) and (1 petameter/ 1015 meters).

We convert 1 exameter into petameters by multiplying 1 exameter by the conversion ratios so that all units other than petameters cancel out: 1 exameter * (1018 meters/ 1 exameter) * (1 petameter/ 1015 meters). Therefore one exameter is 1018/1015  petameters. Finally, when dividing terms with common bases, we subtract the exponents, so our result is 1018/1015 = 103 = 1000.

### Example Question #67 : Proportion / Ratio / Rate

If Shaquille O'Neal is 7 feet tall and casts a shadow 5 feet long. At the same time of day, how long would the shadow be from a 49-foot tall house?

35

45

25

55

15

35

Explanation:

The question is about similar triangles. The height of the two objects would correspond with each other and the shadows would correspond with each other. As with many geometry problems, it is helpful if you draw a diagram.  Set up a proportion for each so the height of Shaquille O'Neal to the height of the house then his shadow to the shadow of the house.

7/49 = 5/x

Solve for x by cross-multiplying:

5 * 49 = 7x

x = 35

### Example Question #68 : Proportion / Ratio / Rate

John is 35 years old, 5 years older than his brother Bob and 20 years younger than his father Jack.  How old was Jack when Bob was born?

20

25

28

5

35

25

Explanation:

If John is 35, that means currently Jack is 55 and Bob is 30.  55 – 30 = 25 years old when Bob was born.

### Example Question #69 : Proportion / Ratio / Rate

In a mixture of flour and sugar, the ratio of flour to sugar is 5 to 1. How many kilograms of flour will there be in 12 kilograms of this mixture?

3

5

1

2

4

2

Explanation:

The question says that the mixture has 5 units of flour for every 1 unit of sugar, which adds up to a total of 5 + 1 = 6 units of the mixture; therefore in 6 kilograms of the mixture, 1 kilogram will be sugar.

To find how much sugar will be in 12 kilograms of the mixture, we multiply the amount of sugar in 6 kilograms of the mixture by 2, giving us 1 kilogram of sugar * 2 = 2 kilograms of sugar.

### Example Question #70 : Proportion / Ratio / Rate

A survey of studio offices in a city with 14,000 employees reveals that there are, on average, 12.5 employees per office. If there have been a cumulative total of 3,400 printers sold to the offices of the city, what is the best estimate of the average number of printers per office?

0.77

3.0

0.33

1.2

4.1