MathematicsPercent as a Ratio Out of One Hundred
Lumi picks up a ratio card showing 3/5 from the desk, holds it next to a large transparent 10×10 grid of one hundred squares, and begins counting aloud how many squares to shade using a cross-hatch pattern — reasoning step by step about how to rescale the denominator to 100 before touching the marker to the grid.
- Explain why a percent is a ratio with a denominator of 100.
- Convert a part-to-whole ratio into a percent by finding an equivalent fraction over 100.
- Identify the percent, part, and whole in everyday contexts such as test scores and sales.
- Calculate a percent when given the part and whole using equivalent ratios.
Key terms
- Percent
- A ratio written as a fraction with a denominator of one hundred, shown with the % sign.
- Ratio
- A comparison of two quantities, such as the part compared to the whole.
- Equivalent fraction
- A fraction that names the same value after multiplying numerator and denominator by the same number.
- Part-to-whole
- A relationship comparing a piece of a group to the entire group it belongs to.
- Scaling factor
- The number you multiply by to rescale a denominator up to one hundred.
Why One Hundred?
Picking a fixed denominator turns every ratio into the same kind of object, so they can be lined up and compared at a glance. One hundred is convenient because our number system is base ten, and a 10 by 10 grid gives exactly one hundred squares. When the denominator is always 100, the numerator alone tells the whole story — 37 out of 100 simply becomes 37 percent.
Converting With Equivalent Fractions
To turn a clean ratio into a percent, find the scaling factor that lifts the denominator to 100, then multiply the numerator by that same factor. For 7 out of 20, ask what times 20 makes 100; the answer is 5, so 7 times 5 gives 35, and the percent is 35. The rule is that whatever you do to the bottom you must also do to the top, which keeps the ratio equivalent.
When The Whole Does Not Divide Evenly
Some denominators, like 12 or 24, do not divide neatly into 100, so the equivalent-fraction trick stalls. In those cases divide the part by the whole to get a decimal, then multiply by 100 to move it onto the percent scale. For 18 out of 24, dividing gives 0.75, and multiplying by 100 gives 75 percent. This division method always works, even when the tidy scaling factor does not exist.
Worked examples
Write 9 out of 20 as a percent.
- Set up the part-to-whole ratio as the fraction 9/20.
- Ask what number times 20 equals 100; since 20 × 5 = 100, the scaling factor is 5.
- Multiply both numerator and denominator by 5: 9 × 5 = 45 and 20 × 5 = 100, giving 45/100.
- A denominator of 100 means the numerator is the percent: 45/100 = 45%.
Answer: 45%
A team won 18 of its 24 games. What percent did they win?
- Write the ratio of games won to games played as 18/24.
- Since 24 does not divide evenly into 100, use the division method: divide the part by the whole, 18 ÷ 24 = 0.75.
- Multiply the decimal by 100 to move onto the percent scale: 0.75 × 100 = 75.
- So the team won 75 out of every 100 games, which is 75%.
Answer: 75%
Activity
Pick up each ratio card, shade the matching squares on the hundred-grid, then drag the correct percent label onto the completed grid.
Practice
Convert the ratio 13 out of 25 students into a percent.
A jar holds 40 marbles and 14 of them are blue. What percent of the marbles are blue?
Common mistakes to avoid
- The numerator is the percentThe numerator is only the part; you must first rescale the whole to 100 before the top number becomes the percent.
- Add to reach one hundredAdding the same number to the top and bottom changes the value; you must multiply both by the same scaling factor instead.
Check your understanding
A class of 25 students has 15 who walk to school. What percent of the class walks to school?
Mira says that 3/5 and 60% are the same value. Is she correct, and why?
Which of the following is the best description of what a percent is?
Recap
A percent is just a ratio rewritten so the whole is one hundred. Find the scaling factor that lifts the denominator to 100 and multiply the numerator by it, or divide the part by the whole and multiply by 100 when the denominator does not cooperate.
Reflect
Where in your own day could you spot a part-to-whole comparison that you could rewrite as a percent?