Practice

Now you’ll have a chance to try a few test-like problems in a scaffolded way. We’ve provided some guidance, but you’ll need to fill in the missing parts of explanations or the step-by-step math to get to the correct answer. Don’t worry—after going through the worked examples at the beginning of this section, these problems should be completely doable.

  1. Ramp meters are often used in and around metropolitan areas to reduce freeway congestion during am and pm rush hours. Depending on freeway volume, ramp meters in Milwaukee allow one car onto the freeway every 5-9 seconds. Assuming a constant ramp car queue, between the hours of 3:30 pm and 6:30 pm, how many more cars can move through a ramp meter with a 5-second interval than one with an 8-second interval?

    1. 270
    2. 320
    3. 810
    4. 960

The following table can help you structure your thinking as you go about solving this problem. Kaplan's strategic thinking is provided, as are bits of structured scratchwork.

Strategic Thinking Math Scratchwork

Step 1: Read the question, identifying and organizing important information as you go

You must determine how many more cars pass through a ramp meter with a 5-second interval.
Step 2: Choose the best strategy to answer the question

One car every 5 (or 8) seconds is a rate, so turn to the DIRT equation. Be careful here; you need to manipulate the given form of the rate before you can use it.


The 3:30 pm to 6:30 pm window translates to 3 hours, which is your time. Your rate, however, involves seconds, so you’ll need to convert time to seconds. Finding d will give you the number of cars; do this for both intervals. Watch your units!


Almost finished. Subtract the 8-second car count from the 5-second car count to find the difference.

5-second interval:

r5 = 1 car per 5 s = ___ cars/s

8-second interval:

r8 = 1 car per 8 s = ___ cars/s

3 h ×____×_____=______s

5 seconds

d5 = ____ × ____

d5 = _____

8 seconds

d8 = ____ × ____

d8 = _____

____ − ____ = ____
Step 3: Check that you answered the right question
If your answer is (C), you’re correct! _____

Here’s another test-like example to try:

  1. Murray’s Annual Income Tax Liability

    Federal
    ($0-$9,225)     		Federal
    ($9,226-$37,450)     		Federal
    ($37,451-$90,750)     		State
    (flat rate)
    10% 15% 25% 4.5%

    Murray has an annual salary of $75,400. He contributes 20% of his pre-tax income to his 401(k), and he pays $150 per month for health insurance (pre-tax, deducted after 401(k)). The table above summarizes Murray’s tax liability; all taxes are calculated based on Murray’s adjusted gross income (pay remaining after 401(k) and insurance payments). He must pay 10% on the first $9,225 in income, 15% on any income between $9,226 and $37,450, 25% on income between $37,451 and $90,750, and a 4.5% state-tax on all of his adjusted gross income. All taxes are deducted simultaneously. What is Murray’s biweekly take-home pay after all deductions have been made?

    1. $1,537.98
    2. $1,586.79
    3. $1,699.78
    4. $1,748.57

The following table can help you structure your thinking as you go about solving this problem. Kaplan's strategic thinking is provided, as are bits of structured scratchwork. If you’re not sure how to approach a question like this, start at the top and work your way down.

Strategic Thinking Math Scratchwork

Step 1: Read the question, identifying and organizing important information as you go
You need to find Murray’s income after deductions. You have information about each deduction, as well as the order in which they’re taken. start ($____) − ____ 401(k)
                             − _____ insurance
                             − taxes
                             = take-home pay
Step 2: Choose the best strategy to answer the question

The table provides tax information for annual income, so don’t convert to biweekly yet. Follow the order you extracted in Step 1 to calculate each deduction first.


To find the amount Murray deducts before taxes, use the three-part percent formula to find his 401(k) contribution, then subtract his health insurance cost.


The question states that all taxes are deducted simultaneously; that is, don’t deduct state tax and then take federal tax on what’s left and vice versa. Use the quantity left after the insurance deduction for all tax calculations, and then subtract Murray’s total tax liability from the remaining quantity post-insurance.





The question asks for Murray’s biweekly take-home pay. Divide by the number of pay periods in one year to get the final answer.

401(k): ___ × ___ = $___ annually

insurance: $___ × ___ = $___ annually

$____  −  $____ −  $____ =  $____ annual pre-tax

taxes: state

_____ × _____ = _____

M. owes $_____ for state

federal 10% bracket

_____ × _____ = _____

M. owes $_____ for 10% bracket

fed 15% bracket

_____ × _____ = _____

M. owes $_____ for 15% bracket

fed 25% bracket

_____ × _____ = _____

M. owes $_____ for 25% bracket

total tax: $_______

take-home: $_____ − $_____

= $_____ annually
Step 3: Check that you answered the right question
If you chose (D), you’re right. _____

Now try your hand at a multi-part question.

  1. Questions 10 and 11 refer to the following information.

  2. Eight concentric squares that share a common center and get progressively larger. The diagram begins with a small gray square in the center and then alternates between black and gray squares. There are a total of four gray squares and four black squares. The outermost square is black.

    Shuang has a set of square ceramic plates she’d like to glaze. She wants to create evenly spaced concentric squares on the plates with gray and black glaze as shown in the diagram above. The squares’ edges are each 0.5 inches apart, and the area of the innermost square is 1 square inch.

  3. What fraction of one plate will Shuang cover with gray glaze?

  4. Shuang also plans to glaze smaller square plates with the same type of pattern as the plate in the figure. The smaller plates’ squares are the same size and distance apart as those of the larger plates. If a small plate has four concentric squares, then the fraction of a small plate that Shuang will cover with black glaze is how many times as great as that of a large plate?

The following table can help you structure your thinking as you go about solving this problem. Kaplan's strategic thinking is provided, as are bits of structured scratchwork. If you’re not sure how to approach a question like this, start at the top and work your way down.

Strategic Thinking Math Scratchwork

Step 1: Read the first question in the set, looking for clues

You’re given a picture of one of Shuang’s plates and told that she will use two colors to create the design.
Step 2: Identify and organize the information you need
You need to find the fraction of the plate that will be gray. gray/total: ?

Step 3: Based on what you know, plan your steps to navigate the first question

To get started, you need to find the side length of the innermost square. Then, you can find the side lengths of the other squares based on that. Label the squares to help keep your calculations clear.
A continuation of the previous image of eight concentric squares. The small gray square in the center is labeled one. The next square, which is black is labeled two. The next square, which is gray, is labeled three, and so on. The outermost square, which is black, is labeled eight.
Step 4: Solve, step-by-step, checking units as you go

You're given that the area of the innermost square is 1 square inch, which means the side length is the square root of that, or 1 inch. Remember how far apart the square edges are; this will help you find the side lengths and areas of the other squares. Also keep in mind you’ll need to account for the fact that each square outside the first is not actually a full square.





To find the ratio of gray glaze to total glaze, divide the gray glaze area by the total glaze area.

sq. 1 = 1 in.2

sq. 2 = ___2  −  ___ = ___ in.2

sq. 3 = ___2  −  ___ = ___ in.2

sq. 4 = ___2  −  ___ = ___ in.2

sq. 5 = ___2  −  ___ = ___ in.2

sq. 6 = ___2  −  ___ = ___ in.2

sq. 7 = ___2  −  ___ = ___ in.2

sq. 8 = ___2  −  ___ = ___ in.2

gray: __ + __ + __ + __ = ___ in.2

black: __ + __ + __ + __ = ___ in.2

gray/total =

Step 5: Did I answer the right question?
If you got 7/16, great job! You’re correct.

Fantastic! Now repeat for the other question in the set. Once again, Kaplan's strategic thinking is provided in the table that follows, as are bits of structured scratchwork. If you’re not sure how to approach the second part, start at the top and work your way down.

Strategic Thinking Math Scratchwork

Step 1: Read the second question in the set, looking for clues
You’re told a smaller plate has four concentric squares with edges 0.5 inches apart as in a larger plate. 4 squares in a small plate

Step 2: Identify and organize the information you need
You’ll need to find the fraction of black glaze on one small plate and one large plate. sm. black: ?
lg. black: ?
Step 3: Based on what you know, plan your steps to navigate the second question
You’ll need to find the black fraction of the two plates. Fortunately, the black fraction for the larger plate is easy to find because you found its gray counterpart in the previous question. large black = 1 −  _____  =  _____
Step 4: Solve, step-by-step, checking units as you go
Good news! To find the black fraction for the small plate, you already did most of the work in the previous question. Just use the same numbers.

square 1 (gray) = ___ in.2

square 2 (black) = ___ in.2

square 3 (gray) = ___ in.2

square 4 (black) = ___ in.2

small gray: ___ + ___ = ___ in.2

small black: ___ + ___ = ___ in.2

small black vs. large black:

___ ÷ ___ = ___ × ___ = ___

Step 5: Did I answer the right question?

Did you get 10/9? If so, congrats! You’re correct.


___