Three faces of one number — and the bridge into ratios and percentages.
Suppose three of the four equal slices of a pie are gone. You can say three‑fourths are gone, or that 0.75 of the pie is gone, or that 75% of it is gone — and every listener pictures exactly the same plate. A fraction, a decimal, and a percent are not three different sizes; they are three spellings of one size. The bar in a fraction was always a division sign, so the fraction quietly hands you the decimal; the place value of a decimal quietly hands you back the fraction; and "out of a hundred" is what turns either one into a percent. This lesson is the hinge: once you can move freely among the three faces, you can grab whichever is handiest for the job — and that habit is the doorway into Stage 4, ratios and percentages.
The horizontal bar in a fraction has been a division sign all along. So 34 simply means 3 ÷ 4. Carry out that division — three wholes shared among four — and you get the decimal 0.75. Nothing new is happening: you are reading the same amount with a different tool.
A fraction is a division waiting to happen: ab = a ÷ b. Divide the top by the bottom and the quotient is the decimal.
Some divisions come out exactly — they terminate. Three‑fourths is one: 3 ÷ 4 = 0.75 and it stops. Others never finish; they repeat forever. One‑third is the classic: 1 ÷ 3 = 0.333… with the 3 going on without end, which we write 0.3 with a bar over the 3. Both are perfectly good decimals — the repeating one just needs the bar to say "and so on."
18 = 1 ÷ 8 = 0.125 (terminates after three places). 27 = 2 ÷ 7 = 0.285714 with a six‑digit block that repeats: 0.(285714) — the same six digits cycle forever.
To go the other way, just listen to how you read the decimal aloud. The number 0.45 reads "forty‑five hundredths," and that sentence is already the fraction: 45100. The last digit sits in the hundredths place, so the bottom is 100. Then reduce — divide top and bottom by their common factor — and you reach lowest terms: 45100 = 920.
The last decimal place names the bottom: one place → tenths (over 10), two places → hundredths (over 100), three places → thousandths (over 1000). Write that fraction, then reduce.
A percent is a fraction with the bottom always fixed at 100. The word says it: per cent means "per hundred." So 75% is shorthand for 75100 — slice the whole into 100 equal parts and take 75 of them. That is exactly the hundredths grid you have been shading: each little cell is one percent, and the count of shaded cells is the percent.
% = per hundred. A percent p is the fraction p100. To turn a fraction into a percent, find its equal name with bottom 100 (or just compute top ÷ bottom × 100).
15 means 1 out of 5; that is 20 out of 100, so 15 = 20%. A test score of 78 works out to 7 ÷ 8 × 100 = 87.5% — a percent need not be a whole number.
Between a decimal and a percent the move is tiny: a percent is just a decimal scaled up by 100. Drop the % sign and shift the point two places left to get the decimal; put the % back and shift two places right to get the percent. So 60% = 0.60 = 0.6, and reading 0.6 by place value gives 610 = 35. One number, three writings: 60% = 0.6 = 35.
The shift is two places, not one. 0.6 = 60%, not 6%. And going the other way, 12 = 0.5 = 50% — not 0.5%. Keep the two‑place shift straight in both directions.
In the wild, the three faces show up wherever they are most convenient, and a fluent reader swaps to whichever one makes the arithmetic easiest. A sign says "25% off" — that is the fraction 14 off, so a quarter of the price comes off. A basketball player's 0.800 hit rate is the same as 80%, which is the same as 45 — four made shots out of every five tried. "3 out of 4" questions correct is 34 = 0.75 = 75%.
A $40 jacket is 25% off. Twenty‑five percent is 14, and a fourth of $40 is $10, so the discount is $10 and you pay $30. Switching to the fraction made the mental math instant.
This freedom to switch is exactly what ratios and percentages build on in the next stage: a percent is a ratio out of a hundred, and "a percent of an amount" is a multiplication you already know. Conversion is the bridge.
| To go from… | …to… | do this | example |
|---|---|---|---|
| fraction | decimal | divide top ÷ bottom | 34 = 3 ÷ 4 = 0.75 |
| decimal | fraction | read the place value, then reduce | 0.45 = 45100 = 920 |
| decimal | percent | shift the point two places right | 0.6 = 60% |
| percent | decimal | shift the point two places left | 75% = 0.75 |
| fraction | percent | top ÷ bottom × 100 | 15 = 20% |
Three faces of one number — a fraction is a division, a decimal is read off place value, and a percent is per hundred.
Write 58 as a decimal, then as a percent.
58 = 5 ÷ 8 = 0.625 (it terminates). As a percent, 0.625 shifts two places to 62.5%.
Turn 0.12 into a fraction in lowest terms.
0.12 is "twelve hundredths," so 12100. Divide top and bottom by 4: 325.
Which is larger, 35 or 0.65? Put both in the same form to decide.
35 = 3 ÷ 5 = 0.6. Compare 0.60 with 0.65: six tenths versus six tenths and five more hundredths, so 0.65 is larger. (As percents: 60% < 65%.)
A quiz is scored "9 out of 10." Give the score as a fraction, a decimal, and a percent.
910 = 9 ÷ 10 = 0.9 = 90%. (Nine of ten equal parts, nine tenths, ninety per hundred.)
A $50 backpack is 20% off. Convert the percent to a fraction and find the dollars saved and the sale price.
20% = 20100 = 15. A fifth of $50 is $10 off, so the sale price is $40.
Write 13 as a decimal. Does it terminate or repeat? Then round it to two places.
1 ÷ 3 = 0.333… — it repeats (written 0.3 with a bar over the 3). The next digit dropped is a 3, which is less than 5, so rounded to two places it is ≈ 0.33.
Six questions to lock it in. Tap the answer you think is right.
This lesson ties together the three written forms of a rational number. It rests on 5.NF.B.3 (interpreting a fraction as the division of numerator by denominator) and 4.NF.C.6 together with the grade‑6 number‑system work (writing decimals as fractions and the reverse), reaches 6.RP.A.3.c (a percent as a rate per hundred and finding a percent of a quantity), and revisits 7.NS.A.2.d (a fraction converts to a terminating or repeating decimal). Encourage your learner to read decimals aloud — "forty‑five hundredths" practically dictates the fraction — and to keep the two‑place shift between decimals and percents firmly in mind. Fluent switching here is the runway into Stage 4, ratios and percentages.