In the name of ALLAH, the most beneficient, the most merciful

Problem Solving

Problem Solving questions test your ability to solve mathematical problems involving arithmetic, algebra, and geometry, as well as word problems, by using problem-solving insight, logic, and the application of basic skills.

The basic skills necessary to do well on this section include high school arithmetic, algebra, and intuitive geometry-no formal trigonometry or calculus is necessary. These skills, along with logical insight into problem-solving situations, are covered by the examination.


Solve each problem in this section by using the information given and your own mathematical calculations. Select the correct answer of the five choices given. Use the scratch paper given for any necessary calculations.

Analysis of Directions

All scratchwork is to be done on the paper given at the test; get used to referring back to the screen as you do your calculations and drawings. You are looking for the one correct answer; therefore, although other answers may be close, there is never more than one right answer.

Suggestions with Examples

Always carefully focus on what you are looking for to ensure that you are answering the right question.

If x + 6 = 9, then 3x + 1 =

  1. 3
  2. 9
  3. 46
  4. 34
  5. 10

You should first focus on 3x + 1, because this is what you are solving for. Solving for x leaves x = 3, and then substituting into 3x + 1 gives 3(3) + 1, or 10. The most common mistake is to solve for x, which is 3, and mistakenly choose A as your answer. But remember, you are solving for 3x + 1, not just x. You should also notice that most of the other choices would all be possible answers if you made common or simple mistakes. The correct answer is C. Make sure that you’re answering the right question.

An employee’s annual salary was increased $15,000. If her new annual salary now equals $90,000, what was the percent increase?

  1. 15%
  2. 16 2⁄3%
  3. 20%
  4. 22%
  5. 24%

Focus on what you are looking for. In this case, percent increase.

Percent increase = change/starting point. If the employee’s salary was increased $15,000 to $90,000, then the starting salary was 90,000 − 15,000 = 75,000. Therefore,

percent increase = 15,000/75,000 = 1/5 = 20%

The correct answer is C.

"Pulling" information out of the word problem structure can often give you a better look at what you are working with, and therefore, you gain additional insight into the problem. Organize this information on your scratch paper.

If a mixture is 3⁄7 alcohol by volume and 4⁄7 water by volume, what is the ratio of the volume of alcohol to the volume of water in this mixture?

  1. 3⁄7
  2. 4⁄7
  3. 3⁄4
  4. 4⁄3
  5. 7⁄4

The first bit of information that you should pull out is what you are looking for: "ratio of the volume of alcohol to the volume of water." Rewrite the ration that you’re looking for as A:W and then rewrite it into its working form: A/W. Next, pull out the volumes of each; A = 3/7 and W = 4/7. Now you can easily figure the answer by inspection or substitution: Using 3⁄7/4⁄7, invert the bottom fraction and multiply to get 3/7 × 7/4 = 3/4. The ratio of the volume of alcohol to the volume of water is 3 to 4. The correct answer is C. When pulling out information, write out the numbers and/or letters on your scratch paper, putting them into some helpful form and eliminating some of the wording.

Sometimes combining terms, performing simple operations, or simplifying the problem in some other way will give you insight and make the problem easier to solve.

Which of the following is equal to 1⁄5 of 0.02 percent?

  1. 0.4
  2. 0.04
  3. 0.004
  4. 0.0004
  5. 0.00004

Simplifying this problem first means changing 1⁄5 to .2. Next change 0.02 percent to 0.0002 (that is, .02 × .01 = 0.0002).

Now that you have simplified the problem, multiply .2 × 0.0002, which gives 0.00004. The correct answer is E. Notice that simplifying can make a problem much easier to solve.

If you immediately recognize the method or proper formula to solve the problem, go ahead and do the work. Work forward.

Which of the following numbers is between 1⁄3 and 1⁄4?

  1. .45
  2. .35
  3. .29
  4. .22
  5. .20

Focus on "between 1⁄3 and 1⁄4." If you know that 1⁄3 is .333 . . . and 1⁄4 is .25, you have insight into the problem and should simply work it forward. Since .29 is the only number between .333 . . . and .25, the correct answer is C. By the way, a quick peek at the answer choices would tip you off that you should work in decimals.

If you don’t immediately recognize a method or formula, or if using the method or formula would take a great deal of time, try working backward - from the answers. Because the answers are usually given in ascending or descending order, almost always start by plugging in choice C first. Then you'll know whether to go up or down with your next try. (Sometimes, you may want to plug in one of the simple answers first.)

If x⁄2 + 3⁄4 = 1 1⁄4, what is the value of x?

  1. −2
  2. −1
  3. 0
  4. 1
  5. 2

You should first focus on “value of x.” If you’ve forgotten how to solve this kind of equation, work backward by plugging in answers. Start with choice C; plug in 0.
0⁄2 + 3⁄4 ≠ 1 1⁄4

Because this answer is too small, try choice D, a larger number. Plugging in 1 gives you
1⁄2 + 3⁄4 = 1 1⁄4
2⁄4 + 3⁄4 = 1 1⁄4
5⁄4 = 1 1⁄4

This answer is true, so D is the correct answer. Working from the answers is a valuable technique.

What is the greatest common factor of the numbers 18, 24, and 30?

  1. 2
  2. 3
  3. 4
  4. 6
  5. 12

The largest number that divides evenly into 18, 24, and 30 is 6. You could've worked from the answers, but here you should start with the largest answer choice, because you’re looking for the greatest common factor. The correct answer is D.

If you don't immediately recognize a method or formula to solve the problem, you may want to try a reasonable approach and then work from the answer choices. Try to be reasonable.

Barney can mow the lawn in 5 hours, and Fred can mow the lawn in 4 hours. How long will it take them to mow the lawn together?

  1. 5 hours
  2. 4 1⁄2 hours
  3. 4 hours
  4. 2 2⁄9 hours
  5. 1 hour

Suppose that you’re unfamiliar with the type of equation for this problem. Try the "reasonable" method. Because Fred can mow the lawn in 4 hours by himself, he will take less than 4 hours if Barney helps him. Therefore, choices A, B, and C are not sensible. Taking this method a little farther, suppose that Barney could also mow the lawn in 4 hours. Therefore, together it would take Barney and Fred 2 hours. But, because Barney is a little slower than this, the total time should be more than 2 hours. The correct answer is D, 2 2⁄9 hours.

Using the equation for this problem would give the following calculations:
1⁄5 + 1⁄4 = 1⁄x

In 1 hour, Barney could do 1⁄5 of the job, and in 1 hour, Fred could do 1⁄4 of the job; unknown 1⁄x is the part of the job they could do together in 1 hour. Now, solving, you calculate as follows:
4⁄20 + 5⁄20 = 1⁄x
9⁄20 = 1⁄x

Cross multiplying gives 9x = 20; therefore, x = 20⁄9, or 2 2⁄9.

Solved Examples
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