One number, three ways
¾, 0.75 and 75% are one amount written three ways. Each way is best at something: fractions for sharing and exact values, decimals for measuring and money, percents for comparing. Moving between them runs through all of Grade 7 — ordering numbers, adding fractions, working out a discount, drawing a circle graph.
Why write one amount three ways?
Because the world has never agreed on one. A recipe says ¾ cup, a kitchen scale shows 0.75 kg, and a sign says 25% off. To compare amounts like these, you need to move between the forms quickly, and to know which form makes a question easy.
Each form is really something different. A fraction is a division: 3/4 means 3 ÷ 4. A decimal is place value: 0.75 is 7 tenths and 5 hundredths. A percent is a fraction with the bottom fixed at 100 — the word comes from the Latin per centum, “for each hundred”. Fixing the bottom is what makes percents easy to compare: 35% and 40% are obviously in order, while 7/20 and 2/5 take work.
Decimals are newer than you might think. In 1585 a Flemish engineer, Simon Stevin, published a short booklet arguing that merchants, surveyors and everyone else should calculate in tenths instead of awkward fractions. The decimal point itself took decades more to settle down.
Where it turns up
- Coins — a quarter is called that because it is ¼ of a dollar: 25 cents, $0.25, or 25% of a dollar
- Nutrition labels — the % Daily Value on a Canadian nutrition facts table compares one serving with a whole day's worth, and Health Canada's rule of thumb is that 5% or less is a little and 15% or more is a lot
- Sales tax — Alberta has no provincial sales tax, so the only one at the till is the 5% GST: 1/20 of the price, or 5 cents on every dollar
- Tools — wrench and drill-bit sizes are often fractions of an inch; a 3/8-inch socket is 0.375 inch, about 9.5 mm, which is why a 10 mm socket is close but not the same
- 1. The words, first
- 2. Converting: three moves
- 3. Benchmarks worth knowing
- 4. Repeating decimals
- 5. Comparing: one form at a time
- 6. The circle-graph angle
The words, first
The idea: Three names, one amount. The last word on the list is where percents turn into angles.
| Word | What it means |
|---|---|
| Fraction | A number written as one whole number over another, like 3/4. It means the top divided by the bottom. |
| Numerator, denominator | The top and bottom of a fraction: in 3/4, 3 pieces of a whole cut into 4. |
| Decimal | A number written with place value past the ones: tenths, hundredths, thousandths. 0.75 is 75 hundredths. |
| Percent (%) | Out of 100. 75% means 75/100. |
| Equivalent | Worth the same. 3/4, 6/8, 0.75 and 75% are all equivalent. |
| Terminating, repeating | A decimal that stops, like 0.375, or one where digits repeat forever, like 0.333… |
| Benchmark | A common amount you know in all three forms, such as ½ = 0.5 = 50%. |
| Central angle | The angle of a slice at the centre of a circle. The whole way round is 360°. |
Converting: three moves
The idea: Fraction to decimal: divide. Decimal to percent: multiply by 100, which moves the point two places right. Percent to fraction: write it over 100 and simplify.
3/8 = 3 ÷ 8 = 0.375 = 37.5% · 45% = 45/100 = 9/20 · 0.6 = 6/10 = 3/5 = 60%
Worked example. Write 7/25 as a decimal and as a percent.
- Can the denominator become 100? 25 × 4 = 100, so multiply the top and bottom by 4: 7/25 = 28/100.
- Read it off: 28 hundredths is 0.28, and 28 out of 100 is 28%.
When the denominator goes into 100, scaling up is quicker than dividing. When it does not, as with 3/8, divide.
Two places, every time. 0.05 is 5%, not 50%. 0.5 is 50%. And 0.375 is 37.5%: the point moves two places, whatever digits happen to be there.
Benchmarks worth knowing
The idea: A handful of amounts come up so often that knowing all three forms by heart saves most of the work in the course.
| Fraction | Decimal | Percent |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 1/10 | 0.1 | 10% |
| 1/8 | 0.125 | 12.5% |
| 1/3 | 0.333… | about 33.3% |
| 2/3 | 0.666… | about 66.7% |
| 1/100 | 0.01 | 1% |
Building percents from 10%. 10% of a number is that number with the point moved one place left. From there, 5% is half of it, 20% is double it, and 1% moves the point two places.
Worked example. A $36 pizza order, and you want to add 15% for the driver.
- 10%: $3.60.
- 5%: half of that, $1.80.
- 15%: 3.60 + 1.80 = $5.40. Check: 0.15 × 36 = 5.4 ✓.
Near a benchmark. 11/20 is just over ½, because ½ is 10/20. So it is a bit more than 50% — it is 55%.
Repeating decimals
The idea: Some fractions never finish when you divide. The digits repeat because the remainders repeat, and a rounded version is close, never equal.
Why 1/3 never stops. Divide 1 by 3. 10 ÷ 3 is 3 with 1 left over, and that leftover 1 puts you back exactly where you started. So the next digit is 3 again, and the next, forever: 1/3 = 0.333…
Rounded is not equal. 0.333… is exactly 1/3. 0.33 is not: it is 33/100, a little less. Three of them make 0.99, not 1.
Patterns to recognise. Ninths repeat their numerator: 1/9 = 0.111…, 2/9 = 0.222…, 5/9 = 0.555… Elevenths repeat 9 times their numerator as a pair: 1/11 = 0.0909…, 2/11 = 0.1818…, 4/11 = 0.3636…
Worked example. Three friends split a $10 bill. Each share is 10/3 = $3.333…, which rounds to $3.33. But 3 × 3.33 = 9.99, so a cent is left over. That missing cent is the repeating decimal that rounding threw away — and why one friend ends up paying $3.34.
Comparing: one form at a time
The idea: Numbers in different forms cannot be compared until they are in the same form. Decimals are usually the quickest to line up.
Worked example. Which is greatest: 5/8, 0.6 or 62%?
- All as decimals: 5/8 = 5 ÷ 8 = 0.625. 62% = 0.62. And 0.6 stays 0.6.
- Same number of places: 0.625, 0.620, 0.600.
- Order: 0.600 < 0.620 < 0.625, so 5/8 is greatest, then 62%, then 0.6.
They are too close to judge by eye, which is exactly when converting pays off.
Percents compare marks out of different totals. 17 out of 20 on one quiz and 21 out of 25 on another: 17/20 = 85/100 = 85%, and 21/25 = 84/100 = 84%. The first was better, just.
The circle-graph angle
The idea: A circle graph is a percent drawn as an angle. The whole circle is 360°, so each slice's angle is its percent of 360°.
25% of 360° = 90° · 10% of 360° = 36° · ⅓ of 360° = 360° ÷ 3 = 120°
Worked example. 14 of the 40 students take the bus. What angle does the bus slice get?
- Fraction: 14/40.
- Decimal: 14 ÷ 40 = 0.35.
- Percent: 35%.
- Angle: 0.35 × 360° = 126°.
Backwards. A slice of 54° is 54/360 = 0.15 of the circle, which is 15% of the whole.
Check the total. The percents must add to 100% and the angles to 360°. If rounding makes them 99% or 361°, find the rounded value before you draw.