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Solving Percent Problems Study Guide

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Updated on Nov 5, 2013

Introduction to Solving Percent Problems

The hardest thing in the world to understand is the income tax.

—ALBERT EINSTEIN, theoretical physicist (1879–1955)

The final percent lesson focuses on another approach to solving percent problems, one that is more direct than the approach described in the previous lesson. It also gives some shortcuts for finding particular percents and teaches how to calculate percent of change (the percent that a figure increases or decreases).

There is a more direct approach to solving percent problems than the shortcut formula you learned in the previous percent lesson:

The direct approach is based on the concept of translating a word problem practically word for word from English statements into mathematical statements. The most important translation rules you'll need are:

  • of means multiply (×)
  • is means equals (=)

You can put this direct approach to work on the three main varieties of percent problems.

Example: What is 15% of 50? (50 is the whole.)

Translation:

  • The word What is the unknown quantity; use the variable w to stand for it.
  • The word is means equals (=).
  • Mathematically, 15% is equivalent to both 0.15 and (your choice, depending on whether you prefer to work in decimals or fractions).
  • of 50 means multiply by 50 (×50).

Put it all together as an equation and solve it:

  w = 0.15 × 50 OR  
  w = 7.5    

Thus, 7.5 (which is the same as ) is 15% of 50.

Tip

Sam's $50 video game is 20% off today. What will the sale price be? There's a shortcut to questions like this. Rather than finding 20% of $50 and then subtracting it from $50, think about it this way: if it's 20% off, then Sam will pay 80% of the game's original price. 80% of $50 is 0.80 × $50 = $40, so the sale price is $40.

Finding What Percent One Number Is of Another Number

Example: 10 is what percent of 40?

Translation:

  • 10 is means 10 is equal to (10 =).
  • What percent is the unknown quantity, so let's use to stand for it. (The variable w is written as a fraction over 100 because the word percent means per 100,or over 100.)
  • of 40 means multiply by 40 (× 40).
Put it all together as an equation and solve:
Write 10 and 40 as fractions:
Multiply fractions:
Reduce:
Cross multiply: 10 × 5 = w × 2
Solve by dividing both sides by 2: 25 = w
Thus, 10 is 25% of 40

 

Tip

Caution! Since the variable w is being written above a 100 denominator, it is being written as a percent and not as a decimal. Therefore, do not move the decimal of your answer—just add the % symbol.

Finding the Whole When a Percent Is Given

Example: 20 is 40% of what number?

Translation:

  • 20 is means 20 equal to (20 =).
  • Mathematically, 40% is equivalent to both 0.40 (which is the same as 0.4) and (which reduces to ). Again, it's your choice, depending on which form you prefer.
  • of what number means multiply by the unknown quantity; let's use w for it (× w).

Put it all together as an equation and solve:

20 = 0.4 ×w OR 20 =
   
   
20 ÷ 0.4 = w ÷ 0.4   20 × 5 = 2 × w
    100 = 2 × w
50 = w   100 = 2 × 50
Thus, 20 is 40% of 50.
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