General Problem-Solving Steps

Strategy 4: Translate from a Figure to an Arithmetic or Algebraic Representation

When a figure is given in a problem, it may be effective to express relationships among the various parts of the figure using arithmetic or algebra.

Sample Question 1 for Strategy 4: Quantitative Comparison Question.

This question is based on the following figure.

Begin figure description.

The figure shows triangle PQR, where P is the leftmost vertex of the horizontal side PR and vertex Q is above PR. Point S lies on horizontal side PR. Point S appears to be the midpoint of PR. Line segment QS is drawn from vertex Q to point S. The lengths of PS and SR appear to be equal.

It is given that the length of PQ is equal to the length of PR.

End figure description.

Quantity A: The length of PS

Quantity B: The length of SR

1. 
   Quantity A is greater.
   
2. 
   Quantity B is greater.
   
3. 
   The two quantities are equal.
   
4. 
   The relationship cannot be determined from the information given.
   

Explanation

From the figure accompanying the question, you know that PQR is a triangle and that point S is between points P and R, so the length of P S is less than the length of P R and the length of SR is less than the length of P R. You are also given that the length of P Q is equal to the length of P R. However, this information is not sufficient to compare the length of P S and the length of S R. Furthermore, because the figure is not necessarily drawn to scale, you cannot determine the relative sizes of the length of P S and the length of S R visually from the figure, though they may appear to be equal. The position of S can vary along side P R anywhere between P and R. Below are two possible variations of the figure accompanying the question, each of which is drawn to be consistent with the information that the length of P Q is equal to the length of P R.

Variation 1
Variation 2

Begin figure description.

In variation 1, instead of appearing to be the midpoint of PR, S appears to be closer to R than to P and the length of PS appears to be greater than the length of SR.

In variation 2, instead of appearing to be the midpoint of PR, S appears to be closer to P than to R and the length of PS appears to be less than the length of SR.

End figure description.

Note that in the previous figures, Quantity A, the length of PS, is greater in Variation 1 and Quantity B, the length of SR, is greater in Variation 2. Thus, the correct answer is Choice D, the relationship cannot be determined from the information given.

Sample Question 2 for Strategy 4: Numeric Entry Question.

This question is based on the following 3-column table, which summarizes the results of a used-car auction. The first row of the table contains column headers. The header for the second column is “Small Cars” and the header for the third column is “Large Cars”. There is no header for the first column. There are 4 rows of data in the table.

Results of a Used-Car Auction

	Small Cars	Large Cars
Number of cars offered	32	23
Number of cars sold	16	20
Projected sales total for cars offered (in thousands)	$70	$150
Actual sales total (in thousands)	$41	$120

For the large cars sold at an auction that is summarized in the table, what was the average sale price per car?

The answer space for this question is preceded by a dollar sign.

Explanation

From the table accompanying the question, you see that the number of large cars sold was 20 and the sales total for large cars was $120,000 (not $120). Thus the average sale price per car was $120,000 over 20 = $6,000. The correct answer is $6,000 (or equivalent).

Strategy 5: Simplify an Arithmetic or Algebraic Representation

Arithmetic and algebraic representations include both expressions and equations. Your facility in simplifying a representation can often lead to a quick solution. Examples include converting from a percent to a decimal, converting from one measurement unit to another, combining like terms in an algebraic expression, and simplifying an equation until its solutions are evident.

Sample Question 1 for Strategy 5: Quantitative Comparison Question.

It is given that y is greater than 4.

Quantity A: the fraction with numerator 3y + 2, and denominator 5

Quantity B: y

1. 
   Quantity A is greater.
   
2. 
   Quantity B is greater.
   
3. 
   The two quantities are equal.
   
4. 
   The relationship cannot be determined from the information given.
   

Explanation

Set up the initial comparison of Quantity A and Quantity B using a placeholder question mark symbol as follows:

the fraction with numerator 3y + 2 and denominator 5, followed by a question mark symbol, followed by y.

Then simplify:

Step 1: Multiply both sides by 5 to get 3y + 2, followed by the question mark symbol, followed by 5y.

Step 2: Subtract 3y from both sides to get 2, followed by the question mark symbol, followed by 2 y.

Step 3: Divide both sides by 2 to get 1, followed by the question mark symbol, followed by y.

The comparison is now simplified as much as possible. In order to compare 1 and y, note that along with Quantities A and B you are given the additional information y is greater than 4. It follows from y is greater than 4 that y is greater than 1 or 1 is less than y, so that in the comparison 1, followed by the question mark symbol, followed by y, the placeholder  represents less than : 1 is less than y.

However, the problem asks for a comparison between Quantity A and Quantity B, not a comparison between 1 and y. To go from the comparison between 1 and y to a comparison between Quantities A and B, start with the last comparison, 1 is less than y, and carefully consider each simplification step in reverse order to determine what each comparison implies about the preceding comparison, all the way back to the comparison between Quantities A and B if possible. Since step 3 was “divide both sides by 2,” multiplying both sides of the comparison 1 is less than y, by 2 implies the preceding comparison 2 is less than 2y, thus reversing step 3. Each simplification step can be reversed as follows:

Step 3 was “Divide both sides by 2.” To reverse this step, you need to multiply both sides by 2. The result of reversing step 3 is 2 is less than 2y.

Step 2 was “Subtract 3y from both sides.” To reverse the step you need to add 3y to both sides. The result of reversing step 2 is 3y + 2 is less than 5y.

Step 1 was “Multiply both sides by 5.” To reverse this step, divide both sides by 5. The result of reversing step 1 is the fraction with numerator 3y + 2 and denominator 5 is less than y.

When each step is reversed, the relationship remains less than , so Quantity A is less than Quantity B. Thus, the correct answer is Choice B, Quantity B is greater.

Sample Question 2 for Strategy 5: Numeric Entry Question.

A merchant made a profit of $5 on the sale of a sweater that cost the merchant $15. What is the profit expressed as a percent of the merchant’s cost?

Give your answer to the nearest whole percent.

The answer space for this question is followed by a % sign.

Explanation

The percent profit is open parenthesis, 5 over 15, close parenthesis, times 100, which equals 33.333..., which equals 33 point 3 with a bar over it, percent, which is 33%, to the nearest whole percent. Thus, the correct answer is 33% (or equivalent).

If you are taking the standard computer-based version of the test, and you use the calculator and the Transfer Display button, the number that will be transferred to the answer box is 33.333333, which is incorrect since it is not given to the nearest whole percent. You will need to adjust the number in the answer box by deleting all of the digits to the right of the decimal point (using the Backspace key).

Also, since you are asked to give the answer as a percent, the decimal equivalent of 33 percent, which is 0.33, is incorrect. The percent symbol next to the answer box indicates that the form of the answer must be a percent. Entering 0.33 in the box would give the erroneous answer 0.33%.