# PSAT Math: Percentages

Percentages aren’t just for test grades; you’ll find them frequently throughout life—discount pricing in stores, income tax brackets, and stock price trackers all use percents in some form. It’s critical that you know how to use them correctly, especially on Test Day.

Suppose you have a bag containing 10 blue marbles and 15 pink marbles, and you’re asked what percent of the marbles are pink. You can determine this easily by using the formula $\mathrm{Percent}=\frac{\text{part}}{\text{whole}}×100\mathrm{%}$. Plug 15 in for the part and 10 + 15 (= 25) for the whole to get $\frac{15}{25}×100\mathrm{%}=60\mathrm{%}$ pink marbles.

Another easy way to solve many percent problems is to use the following statement: (blank) percent of (blank) is (blank). Translating from English into math, you obtain (blank)% × (blank) = (blank). As you saw with the DIRT equation in the rates section, knowledge of any two quantities will unlock the third.

You might also be asked to determine the percent change in a given situation. Fortunately, you can find this easily using a variant of the percent formula:

Sometimes more than one change will occur. Be especially careful here, as it can be tempting to take a “shortcut” by just adding two percent changes together (which will almost always lead to an incorrect answer). Instead you’ll need to find the total amount of the increase or decrease and calculate accordingly.

An example of a multi-part question that tests your percentage expertise follows.

Work through the Kaplan Method for Multi-Part Math Questions step-by-step to solve this question. The following table shows Kaplan’s strategic thinking on the left, along with suggested math scratchwork on the right.

 Strategic Thinking Math Scratchwork Step 1: Read the first question in the set, looking for clues   The intro provides information on two account types. regular acct: 0.25%, \$5,000 min   student acct: 0.42%, \$1,000 min Step 2: Identify and organize the information you need   You need to find how much more interest the \$5,000 account will have after three years. difference in interest: ? Step 3: Based on what you know, plan your steps to navigate the first question   What pieces needed to find the answer are missing? How do you find the difference in interest?You’ll need the amount of interest that each account accrues after three years. Use the three-part percent formula to find annual interest, then find the interest after three years, then take the difference. reg. int. = ?   stu. int. = ?   reg. int. x 3 = ?   stu. int. x 3 = ? reg. – stu. = ? (blank)% of (blank) is (blank) Step 4: Solve, step-by-step, checking units as you go   How much interest does each account earn after one year? After three years?Plug in appropriate values. Remember to adjust the decimal point on the percents appropriately. Triple the interest amounts to get the total accrued interest after three years.What’s the difference in interest earned? Subtract. 0.0025 x \$5,000 = \$12.50   0.0042 x \$1,000 = \$4.20   \$12.50 x 3 = \$37.50   \$4.20 x 3 = \$12.60 \$37.50 – \$12.60 = \$24.90 Step 5: Did I answer the right question?   You’ve found how much more interest the regular account makes after three years, so you’re done with the first question. 24.9

The first part of the question set is finished! Now on to Step 6: Repeat for the other question in the set.

The following table shows Kaplan’s strategic thinking on the left, along with suggested math scratchwork on the right.

Strategic Thinking Math Scratchwork

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

No new information here, but some pieces from the first part of the question set might be useful.

Step 2: Identify and organize the information you need

What does the second question ask you to find? Is there any information from the first question that will help?

The second question asks for the student account balance that will yield the same interest as a regular account at the minimum balance. You know the interest rates for a regular account and a student account at their respective minimum balances. You also know the interest earned annually (from the first question).

reg. acct: 0.25%, \$12.50/yr
stu. acct: 0.42%,
\$4.20/yr

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

How can you determine when the two accounts will earn the same interest? Will algebra work here?

To answer the second question, you’ll need to find when annual student interest equals annual regular interest. Set up equations with interest earned as a function of the account balance, one for each account. You already know what the regular account makes in interest annually, so it’s just a matter of finding when the student equation equals that value.

y = mx + b

Step 4: Solve, step-by-step, checking units as you go

When does the student account earn \$12.50 in interest per year?

Plug 12.5 in for your dependent variable, then solve for x.

reg.: y = 0.0025x
12.5 = 0.0025 x 5,000

stu.: y = 0.0042x
12.5 = 0.0042x

x = 2,976.19

Step 5: Did I answer the right question?

The second question asks for the balance a student account needs to make the same interest as a regular account. Round to the nearest dollar, and you’re done!

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