Converting among fractions, decimals and percents
Three fifths of the seeds sprouted, 0.6 of them sprouted, 60\% of them sprouted: all three say the same thing. How do you turn any one of these forms into the others, and what happens when the division never comes out even?
Where you are 8 in Percent
When you reach for it
A seed packet from Bao’s seed bank promises that 60\% of its seeds will sprout. The bank’s test tray records that 0.6 of the seeds sprouted, and a grower writes that three fifths came up. All three say the same thing in different spellings: a percent, a decimal and a fraction are three names for one number.
You reach for a conversion whenever the name you have is not the one you need. Two amounts given in different forms cannot be set side by side until they share one, and putting them in order is Comparing and ordering rational numbers. A percent has to become a decimal or a fraction before you can multiply with it, which is how a percent of a quantity is found. A fraction such as \displaystyle \frac{5}{16} is exact but hard to picture, while 31.25\% says at once that it is a little under a third. And a ratio between two parts, such as “3 tire jobs for every 5 gear jobs”, often has to become a share of the whole.
The procedure
Every conversion runs through one fact from What percent means: a rate per hundred: a percent is a number of hundredths. 60\% is sixty hundredths, \displaystyle \frac{60}{100}, and sixty hundredths written as a decimal is 0.60, or 0.6. So changing the name means writing the amount as hundredths and reading it off in the form you want. Figure 1 lines up the three names on one number line from 0 to 1, the whole, which is 100\%.
Part 1: three spellings of hundredths
A percent as a fraction. Write the percent as that many hundredths, then simplify. 24\% is \displaystyle \frac{24}{100}, and dividing the top and the bottom by their common factor 4 gives an equivalent fraction:
A fraction is in simplest form when its top and bottom share no factor except 1, and a fraction answer is given that way. When the percent has digits after the point, the hundredths fraction has a decimal on top, and multiplying top and bottom by 10 clears it (Example 1).
A percent as a decimal. Sixty hundredths is 0.60 because the hundredths place is the second place after the point. So write the percent’s number with its ones digit in the hundredths place: 24\% = 0.24, 5\% = 0.05 and 130\% = 1.30 = 1.3. Every digit has moved two places to the right, because every digit has been divided by 100; since the point stays where it is, it looks as if the point moved two places left. The 0 in 0.05 matters: 0.5 would be fifty hundredths.
A decimal as a percent. Run the same move backwards and count hundredths. 0.7 is 0.70, seventy hundredths, so 0.7 = 70\%, and 0.042 is 4.2 hundredths, so 0.042 = 4.2\%. Every digit moves two places to the left, because one whole is 100 hundredths, 1 = 100\%, so the number of hundredths in an amount is 100 times the amount.
A fraction as a percent. The fraction has to become hundredths, and there are two ways.
Scale to hundredths. If the bottom can be multiplied or divided to reach 100, an equivalent fraction does the job. For \displaystyle \frac{3}{4}, multiply top and bottom by 25, since 4 \times 25 = 100:
This works when the bottom is a factor of 100 (2, 4, 5, 10, 20, 25 or 50), or a number such as 300 or 1000 that divides down to 100 (Example 3).
Divide. The fraction bar means division, so \displaystyle \frac{3}{4} is the number 3 \div 4. Long division gives the decimal 0.75, as in Converting between fractions and decimals, and the decimal becomes the percent 75\%. Division works for every fraction, including those whose bottom never reaches 100, such as \displaystyle \frac{5}{16} (Example 4).
The two ways agree because both find how many hundredths the fraction is worth.
Part 2: when the division never ends
Try \displaystyle \frac{1}{3}. No whole number times 3 makes 100, so scaling fails. Division gives this:
| Place after the point | Divide | Digit | Remainder |
|---|---|---|---|
| first | 10 \div 3 | 3 | 1 |
| second | 10 \div 3 | 3 | 1 |
| third | 10 \div 3 | 3 | 1 |
The remainder 1, with a 0 brought down, is 10 again, so the next step is the same step, and so is every step after it. The threes never stop: \displaystyle \frac{1}{3} = 0.333\ldots, written 0.\overline{3} with a bar over the digit that repeats (Terminating and repeating decimals).
So \displaystyle \frac{1}{3} is not a whole number of hundredths, and no decimal that ends is exactly equal to it. The percent can still be exact if it has a fraction part. A percent counts shaded squares on a hundred grid, so \displaystyle \frac{1}{3} shades a third of 100 squares: 100 \div 3 = 33 remainder 1, which is 33 whole squares and one more square shared three ways.
The decimal 33.333\ldots\%, or 33.\overline{3}\%, names the same amount. 33.3\% is that number rounded to the nearest tenth of a percent: close, but not equal.
The grid gives a way that works for every fraction: the percent is that fraction of 100, so multiply the top by 100 and divide by the bottom. For \displaystyle \frac{1}{3} that is 100 \div 3, and for \displaystyle \frac{3}{4} it is 300 \div 4 = 75.
Which fractions repeat. In simplest form, a fraction’s decimal ends exactly when its bottom can be multiplied up to 10, 100, 1000 or another power of ten. Those numbers are built from 2s and 5s only (1000 = 2 \times 2 \times 2 \times 5 \times 5 \times 5), so a bottom with any other prime factor, such as 3, 7 or 11, can never reach one, and its decimal repeats. It must repeat rather than wander: in division by 6 the only possible remainders are 0 to 5, so a remainder soon comes round again, and from there the steps, and the digits, run as before.
Back from a percent with a fraction part. \displaystyle 33\frac{1}{3}\% is \displaystyle 33\frac{1}{3} hundredths. Multiplying top and bottom by 3 clears the third, because \displaystyle 33\frac{1}{3} \times 3 = 100:
Fractions, ratios and percents as one idea
A Part-to-part and part-to-whole ratios compares two parts of one whole. If a seed mix holds sunflower and marigold seeds in the ratio 2 : 3, then every 5 seeds hold 2 sunflower seeds and 3 marigold seeds. Sunflower seeds are \displaystyle \frac{2}{5} of the mix, which is \displaystyle \frac{40}{100}, or 40\%, and marigold seeds are the other 60\%.
The ratio 2 : 3, the fraction \displaystyle \frac{2}{5} and the percent 40\% describe one mix. The fraction and the percent compare a part with the whole; the ratio compares a part with a part, and the whole is found by adding the parts. The ratio also gives one part as a percent of the other: sunflower seeds number \displaystyle \frac{2}{3} of the marigold seeds, \displaystyle 66\frac{2}{3}\% of them. Both percents are right. They answer different questions because they measure against different wholes (Example 7).
Worked examples
Examples 1 to 4 go with Part 1, and Examples 5 to 7 with Part 2.
Percents from a repair shop
Aditi’s bike repair shop records that last month 45\% of its jobs were punctures and 8\% were broken chains, and that 62.5\% of the bikes brought in were children’s bikes. Write each percent as a decimal and as a fraction in simplest form.
Solution45\% is forty-five hundredths. As a decimal the 5 sits in the hundredths place: 0.45. As a fraction it is \displaystyle \frac{45}{100}, and 45 and 100 share the factor 5:
8\% is eight hundredths. The 8 belongs in the hundredths place, so a 0 holds the tenths place: 0.08, not 0.8. As a fraction, \displaystyle \frac{8}{100} simplifies by 4:
62.5\% is 62.5 hundredths. As a decimal the 2 lands in the hundredths place and the 5 in the thousandths: 0.625. As a fraction, \displaystyle \frac{62.5}{100} has a decimal on top, so multiply top and bottom by 10 to get whole numbers, then divide both by 125, the largest factor they share:
Check. 5 \div 8 = 0.625, the same decimal.
Germination rates at the seed bank
Bao’s seed bank tests old seed by planting a tray of each kind. Write each share of seeds that sprouted as a percent: (a) 0.9 of the tomato seeds, (b) 0.07 of the parsnip seeds, (c) 0.355 of the onion seeds.
SolutionCount hundredths in each.
(a) 0.9 = 0.90, ninety hundredths, so 90\% of the tomato seeds sprouted. The extra 0 changes nothing about the amount; it only makes the hundredths visible.
(b) 0.07 is seven hundredths, so 7\%. The digit 7 sits in the hundredths place, and moving it two places left brings it to the ones place of 7.
(c) 0.355 is 35.5 hundredths, since 3 tenths and 5 hundredths are 35 hundredths and the last 5 thousandths are half of one more hundredth. So 35.5\% of the onion seeds sprouted.
In each case the digits moved two places to the left, which is multiplying by 100.
How much of each puzzle is done
Cesar has placed 640 pieces of a 1000-piece jigsaw, Dafne 215 pieces of a 500-piece one, and Eitan 78 pieces of a 300-piece one. What percent of each puzzle is done?
SolutionEach fraction placed can be scaled to hundredths by dividing top and bottom by the same number, which gives an equivalent fraction.
Cesar: 1000 \div 10 = 100, so divide both by 10:
Dafne: 500 \div 5 = 100, so divide both by 5:
Eitan: 300 \div 3 = 100, so divide both by 3:
The counts 640, 215 and 78 are pieces of puzzles of different sizes. The percents measure each against a whole of 100, so they say how far through each puzzle its builder is.
When the bottom does not reach a hundred
In Bao’s seed bank, 5 of the 16 jars of bean seed hold runner beans, and 3 of the 40 packets in a box were damaged by damp. Write each as a percent.
SolutionNo whole number times 16 makes 100, so divide 5 by 16:
| Place after the point | Divide | Digit | Remainder |
|---|---|---|---|
| first | 50 \div 16 | 3 | 2 |
| second | 20 \div 16 | 1 | 4 |
| third | 40 \div 16 | 2 | 8 |
| fourth | 80 \div 16 | 5 | 0 |
The remainder reached 0, so the decimal ends: \displaystyle \frac{5}{16} = 0.3125. That is 31.25 hundredths, so 31.25\% of the jars hold runner beans.
For the packets, 40 does not go into 100 a whole number of times either, but 40 \times 25 = 1000:
Division gives the same: 3 \div 40 = 0.075. A share of 7.5\% means 7.5 packets for every 100, which is 15 for every 200.
Brake pads in thirds
At Aditi’s bike repair shop, 1 in every 3 bikes brought in needs new brake pads and the other 2 in 3 do not. Write \displaystyle \frac{1}{3} and \displaystyle \frac{2}{3} as decimals and as exact percents, and then as percents rounded to the nearest tenth.
SolutionDividing 1 by 3 gives remainder 1 at every step, as in the table above, so \displaystyle \frac{1}{3} = 0.\overline{3}. Dividing 2 by 3 gives 20 \div 3 = 6 remainder 2 at every step, so \displaystyle \frac{2}{3} = 0.\overline{6}.
For the exact percents, take each fraction of 100. 100 \div 3 = 33 remainder 1, and that 1 shared three ways is \displaystyle \frac{1}{3}. 200 \div 3 = 66 remainder 2, and 2 shared three ways is \displaystyle \frac{2}{3}:
Check. The two groups make up all the bikes, and \displaystyle 33\frac{1}{3}\% + 66\frac{2}{3}\% = 100\%.
Rounded to the nearest tenth, 33.333\ldots\% is 33.3\%, since the next digit is 3, and 66.666\ldots\% is 66.7\%, since the next digit is 6. One rounds down and one rounds up, and they still add to 100.0\%.
A repeat that starts late, and the way back
(a) Dafne has finished \displaystyle \frac{5}{6} of a puzzle. Write this as a decimal and as a percent.
(b) \displaystyle 58\frac{1}{3}\% of the pieces of Cesar’s puzzle are sky. What fraction of the pieces is that, in simplest form?
Solution(a) Divide 5 by 6. First 50 \div 6 = 8 remainder 2, then 20 \div 6 = 3 remainder 2, and the remainder 2 comes round again, so every later digit is 3:
The 8 does not repeat, so the bar sits over the 3 alone. For the exact percent, take \displaystyle \frac{5}{6} of 100: 500 \div 6 = 83 remainder 2, and \displaystyle \frac{2}{6} = \frac{1}{3}. So \displaystyle \frac{5}{6} = 83\frac{1}{3}\%, which agrees with the decimal 0.8333\ldots, or 83.333\ldots hundredths.
(b) \displaystyle 58\frac{1}{3}\% is \displaystyle 58\frac{1}{3} hundredths. Multiply top and bottom by 3 to clear the third: 58 \times 3 = 174, and the third makes 1 more, so the top is 175. Then divide top and bottom by their common factor 25:
Check. 7 \div 12 = 0.58333\ldots, which is 58.333\ldots hundredths.
Three kinds of job
Aditi sorts last month’s jobs into tire jobs, brake jobs and gear jobs, in the ratio 3 : 4 : 5. What fraction and what percent of all the jobs is each kind? What percent of the number of gear jobs is the number of tire jobs?
The ratio says that every 3 + 4 + 5 = 12 jobs include 3 tire jobs, 4 brake jobs and 5 gear jobs. So the whole is 12 equal units, as on the tape, and each kind’s fraction of the whole is its number of units over 12:
As percents: \displaystyle \frac{1}{4} = \frac{25}{100} = 25\%, and \displaystyle \frac{1}{3} = 33\frac{1}{3}\% as in Example 5. For \displaystyle \frac{5}{12}, take it of 100: 500 \div 12 = 41 remainder 8, and \displaystyle \frac{8}{12} = \frac{2}{3}, so \displaystyle \frac{5}{12} = 41\frac{2}{3}\%.
Check. \displaystyle 25\% + 33\frac{1}{3}\% + 41\frac{2}{3}\% = 100\%, all the jobs.
The last question measures against a different whole: not all the jobs, only the gear jobs. Tire jobs to gear jobs is 3 : 5, so the tire jobs number \displaystyle \frac{3}{5} of the gear jobs, which is \displaystyle \frac{60}{100}, or 60\%. Tire jobs are 25\% of all jobs and 60\% of the gear jobs, and both are true, because “of what” changed.
Boundaries
A percent and its fraction share one whole. 25\% of the jobs and \displaystyle \frac{1}{4} of the jobs are the same amount; 25\% of the jobs and 25\% of the gear jobs are not. Converting changes the name of a share, never the whole it is measured against.
A rounded percent is not the fraction. \displaystyle \frac{1}{3} is \displaystyle 33\frac{1}{3}\% exactly, while 33.3\% falls short: 33.3\% of 3000 seeds is 999 seeds, but a third of them is 1000. Write the fraction part, or the repeating bar, when the exact value matters.
Place value decides the zeros. The “two places” rule is about place value, not about counting digits on the page. 5\% is 0.05, and 0.5 is 50\%.
More than the whole. A fraction greater than 1 converts the same way and gives a percent above 100\%: \displaystyle \frac{5}{4} = 1.25 = 125\%. Percents above 100 and below 1 is about what such percents mean.
A ratio of parts is not a fraction of the whole. In the ratio 2 : 3 the first part is \displaystyle \frac{2}{5} of the whole, not \displaystyle \frac{2}{3}; \displaystyle \frac{2}{3} compares it with the other part.
A fraction answer is finished in simplest form. \displaystyle \frac{45}{100} and \displaystyle \frac{9}{20} are equal, but the simplest form shows the comparison with the smallest numbers.
Where this goes next
With any amount able to wear any of its three names, Finding a percent of a quantity turns a percent into a decimal to find a percent of a quantity, and Benchmark percents and mental percent learns the friendliest conversions, 50\%, 25\% and 10\%, well enough to work in your head. Comparing and ordering rational numbers puts fractions, decimals and percents on one line, and Finding what percent one number is of another turns a part and a whole into a percent, which is this lesson’s fraction-to-percent move with the fraction built from a situation; the three percent questions gather all of these together. In Basic probability a chance is written as a fraction, a decimal or a percent, whichever reads best.
The repeating decimals of Part 2 go further. Turning a repeating decimal into a fraction runs the long division backwards and turns any repeating decimal into a fraction, which is why every rational number has a decimal that either ends or repeats. And \displaystyle 0.333\ldots = \frac{3}{10} + \frac{3}{100} + \frac{3}{1000} + \ldots is an endless sum that adds up to exactly \displaystyle \frac{1}{3}, the first example of an Infinite geometric series.
Exercises
Part 1: percents as decimals and fractions
Write each percent as a decimal and as a fraction in simplest form.
- 35\%
- 4\%
- 2.5\%
- 140\%
Part 1: decimals as percents
- In a seed bank’s test, 0.3 of the carrot seeds sprouted. What percent is that?
- Write 0.06 as a percent.
- Write 0.825 as a percent.
Part 1: fractions as percents
- Write \displaystyle \frac{19}{25} as a percent.
- Write \displaystyle \frac{9}{50} as a percent.
- In a seed bank, 156 of 400 jars hold flower seeds. What percent of the jars is that?
- Write \displaystyle \frac{7}{16} as a decimal and as a percent.
- Write \displaystyle \frac{9}{40} as a decimal and as a percent.
Part 2: fractions whose decimals repeat
Write each fraction as a decimal, using a bar, and as an exact percent.
- \displaystyle \frac{1}{6}
- \displaystyle \frac{4}{9}
- \displaystyle \frac{11}{12}
- \displaystyle \frac{3}{11}, and give the percent rounded to the nearest tenth as well.
Part 2: back to fractions
Write each percent as a fraction in simplest form.
- \displaystyle 8\frac{1}{3}\%
- \displaystyle 22\frac{2}{9}\%
Part 2: ratios as fractions and percents
- In a jigsaw, edge pieces and inner pieces are in the ratio 7 : 43. What percent of the pieces are edge pieces, and what percent are inner pieces?
- A seed mix holds bean, pea and squash seeds in the ratio 1 : 3 : 4. Write each kind as a fraction of the mix and as a percent.
- A bike repair shop has 4 children’s bikes and 16 adult bikes waiting. What percent of the number of adult bikes is the number of children’s bikes? What percent of all the waiting bikes are children’s bikes?
Reasoning
- Explain why writing a decimal as a percent moves each digit two places to the left, using hundredths.
- Explain why the long division of 1 by 3 never ends, and why 33.3\% of a quantity is not exactly a third of it.
- Without dividing, decide which of \displaystyle \frac{7}{40} and \displaystyle \frac{7}{30} has a decimal that repeats. Explain.
Answers
- 0.35 and \displaystyle \frac{7}{20}. Thirty-five hundredths; \displaystyle \frac{35}{100} simplifies by 5.
- 0.04 and \displaystyle \frac{1}{25}. Four hundredths, so the 4 sits in the hundredths place; \displaystyle \frac{4}{100} simplifies by 4.
- 0.025 and \displaystyle \frac{1}{40}. \displaystyle \frac{2.5}{100} = \frac{25}{1000}, which simplifies by 25.
- 1.4 and \displaystyle \frac{7}{5}. \displaystyle \frac{140}{100} simplifies by 20; it is more than one whole.
- 30\%. 0.3 = 0.30, thirty hundredths.
- 6\%. Six hundredths.
- 82.5\%. 0.825 is 82.5 hundredths.
- 76\%. Multiply top and bottom by 4: \displaystyle \frac{76}{100}.
- 18\%. Multiply top and bottom by 2: \displaystyle \frac{18}{100}.
- 39\%. Divide top and bottom by 4: \displaystyle \frac{156}{400} = \frac{39}{100}.
- 0.4375 and 43.75\%. Long division of 7 by 16 gives the digits 4, 3, 7, 5, with remainders 6, 12, 8 and 0.
- 0.225 and 22.5\%. 40 \times 25 = 1000, so \displaystyle \frac{9}{40} = \frac{225}{1000} = 0.225.
- 0.1\overline{6} and \displaystyle 16\frac{2}{3}\%. 10 \div 6 = 1 remainder 4, then 40 \div 6 = 6 remainder 4 for ever. Of 100: 100 \div 6 = 16 remainder 4, and \displaystyle \frac{4}{6} = \frac{2}{3}.
- 0.\overline{4} and \displaystyle 44\frac{4}{9}\%. Each step is 40 \div 9 = 4 remainder 4. Of 100: 400 \div 9 = 44 remainder 4.
- 0.91\overline{6} and \displaystyle 91\frac{2}{3}\%. The digits are 9 and 1, then 80 \div 12 = 6 remainder 8 for ever. Of 100: 1100 \div 12 = 91 remainder 8, and \displaystyle \frac{8}{12} = \frac{2}{3}.
- 0.\overline{27}, \displaystyle 27\frac{3}{11}\%, and 27.3\% rounded. The remainders alternate 8, 3, 8, 3, giving 2, 7, 2, 7. Of 100: 300 \div 11 = 27 remainder 3. The percent 27.2727\ldots\% rounds up to 27.3\%.
- \displaystyle \frac{1}{12}. Multiply top and bottom by 3: \displaystyle \frac{25}{300}, which simplifies by 25.
- \displaystyle \frac{2}{9}. Multiply top and bottom by 9: 22 \times 9 + 2 = 200, so \displaystyle \frac{200}{900}, which simplifies by 100.
- 14\% edge and 86\% inner. The whole is 7 + 43 = 50 units: \displaystyle \frac{7}{50} = \frac{14}{100} and \displaystyle \frac{43}{50} = \frac{86}{100}.
- Bean \displaystyle \frac{1}{8} = 12.5\%, pea \displaystyle \frac{3}{8} = 37.5\%, squash \displaystyle \frac{1}{2} = 50\%. The whole is 1 + 3 + 4 = 8 units, and 1 \div 8 = 0.125; squash is \displaystyle \frac{4}{8}. The three add to 100\%.
- 25\% of the adult bikes, and 20\% of all the bikes. \displaystyle \frac{4}{16} = \frac{1}{4}, but all the bikes number 4 + 16 = 20, and \displaystyle \frac{4}{20} = \frac{1}{5} = \frac{20}{100}.
- A percent counts hundredths. One whole is 100 hundredths, a tenth is 10 hundredths and a hundredth is 1 hundredth, so the number of hundredths in a decimal is 100 times the decimal. Multiplying by 100 makes every digit worth 100 times as much, which moves it two places to the left: the tenths digit becomes the tens digit of the percent.
- Each step of 1 \div 3 leaves remainder 1, and with a 0 brought down that is 10 again, so the same step repeats for ever and the remainder never reaches 0. Since \displaystyle \frac{1}{3} is \displaystyle 33\frac{1}{3}\%, the rounded 33.3\% is \displaystyle \frac{1}{30} of a percent short, and of a large enough quantity that gap is visible: 33.3\% of 300 is 99.9, not 100.
- \displaystyle \frac{7}{30} repeats. Both are in simplest form, since 7 shares no factor with 40 or 30. 40 = 2 \times 2 \times 2 \times 5 has only 2s and 5s, so 40 \times 25 = 1000 and \displaystyle \frac{7}{40} = 0.175 ends. 30 = 2 \times 3 \times 5 has the factor 3, and no multiple of 30 by a whole number can be a power of ten, which has only 2s and 5s, so its decimal repeats (0.2\overline{3}).