### All PSAT Math Resources

## Example Questions

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

The price of 10 yards of fabric is *c* cents, and each yard makes *q* quilts. In terms of *q** *and *c*, what is the cost, in cents, of the fabric required to make 1 quilt?

**Possible Answers:**

10cq

(10c )/(q )

(cq )/(10 )

(c )/(10q )

**Correct answer:**

(c )/(10q )

We create a conversion ratio that causes yards to cancel out, leaving only cents in the numerator and quilts in the denominator. This ratio is ((c cent )/(10 yard))((1 yard)/(q quilt))=(c )/(10q ) cent⁄quilt . Since the ratio has cents in the numerator and quilts in the denominator, it represents the price in cents per quilt.

### Example Question #1 : How To Find Proportion

Susan is doing a bake sale for her sorority. One third of the money she made is from blueberry cupcakes, which cost 50 cents each. A quarter of her sales is from cinnamon cream pies, which cost $1 each. And the rest are from her chocolate brownies, which cost 25 cents each. She made a total of $60 at the end of her bake sale, how many brownies did she sell?

**Possible Answers:**

140

100

120

150

130

**Correct answer:**

100

1/3 of sales from cupcakes = $20, ¼ of sales from cream pies = $15 and the rest are from brownies = $60-$20-$15 = $25. Since each brownie costs 25 cents, Susan will have sold 100 of them.

### Example Question #3 : How To Find Proportion

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?

**Possible Answers:**

8

6

5

9

12

**Correct answer:**

8

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 : Arithmetic

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?

**Possible Answers:**

3

2

5

12

15

**Correct answer:**

5

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 #1 : How To Find Proportion

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:**

$1794.87

$1060.60

$2855.47

$2692.31

$3181.81

**Correct answer:**

$2855.47

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 #1 : How To Find Proportion

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?

**Possible Answers:**

8

6

4

5

2

**Correct answer:**

2

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 #2 : How To Find Proportion

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

**Possible Answers:**

Quantity A is greater

The two quantities are equal

Quantity B is greater

The relationship cannot be determined from the information given

**Correct answer:**

Quantity B is greater

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 #2 : How To Find Proportion

If 10^{15} meters = 1 petameter and 10^{18} meters = 1 exameter, how many petameters are equal to 1 exameter?

**Possible Answers:**

**Correct answer:**1000

The problem gives us two conversion ratios, which are (10^{18} meters/ 1 exameter) and (1 petameter/ 10^{15} 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 * (10^{18} meters/ 1 exameter) * (1 petameter/ 10^{15} meters). Therefore one exameter is 10^{18}_{/10}^{15 }petameters. Finally, when dividing terms with common bases, we subtract the exponents, so our result is 10^{18}_{/10}^{15 }= 10^{3} = 1000.

### Example Question #1621 : Psat Mathematics

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?

**Possible Answers:**

25

15

55

45

35

**Correct answer:**

35

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 #61 : Fractions

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?

**Possible Answers:**

28

25

35

20

5

**Correct answer:**

25

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

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