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Part-to-part and part-to-whole ratios

Two red flags for every five white ones. Does that mean two fifths of the flags are red?

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What a ratio is: language and notationMath 6
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Tape diagrams and sharing in a ratioMath 6What percent means: a rate per hundredMath 6
Where you are 3 in What a ratio is
Fractions, ratio and percent3
Ratio6
What a ratio is3
Deciding which of two ratios is larger
Part-to-part and part-to-whole ratios
What a ratio is: language and notation

Why this exists

Marisol is hanging bunting for a party: 2 red flags, then 5 white ones, then 2 red and 5 white again, all the way along. “Two to five” describes the pattern. But two to five of what?

Ask what part of the bunting is red, and \displaystyle \frac{2}{5} is the wrong answer. Each repeat of the pattern has 7 flags, 2 of them red, so the red part of the string is \displaystyle \frac{2}{7}.

A ratio can compare a part with another part (red flags with white flags) or a part with the whole (red flags with all the flags). Both use a colon, and for the same string they give different numbers. This lesson names the two kinds, moves between them, extends the idea to three or more parts, and pins down when a ratio says the same thing as a fraction.

Definition

Take a collection split into parts, so that every item belongs to exactly one part. How a ratio is written and read is in What a ratio is: language and notation; here the question is what is compared.

  • A part-to-part ratio compares one part with another part. Red flags to white flags is 2 : 5.
  • A part-to-whole ratio compares one part with the whole collection. Red flags to all flags is 2 : 7.
  • The whole is all the parts together, so its number is the sum of the parts’ numbers: 2 + 5 = 7.

A ratio of three quantities, written 2 : 3 : 4 and read “two to three to four”, says that for every 2 of the first there are 3 of the second and 4 of the third, all at once. Any two of its numbers, kept in order, compare those two parts, and the sum 2 + 3 + 4 = 9 stands for the whole. Four or more parts work the same way.

A part-to-whole ratio and a fraction say the same thing: red to all flags 2 : 7 means \displaystyle \frac{2}{7} of the flags are red, a fraction of a whole. A part-to-part ratio does not name a fraction of the collection: 2 : 5 does not mean \displaystyle \frac{2}{5} of the flags are red. That \displaystyle \frac{2}{5} compares red with white only: the number of red flags is \displaystyle \frac{2}{5} of the number of white flags.

Intuition

Felix collects pebbles on a beach. Every handful he picks up has 4 gray pebbles and 3 white ones. Figure 1 draws each handful as a column.

4 gray 3 white
Figure 1Felix’s pebbles, one column for each handful. Every column holds 4 gray and 3 white.

One picture holds four ratios:

ComparisonKindRatio
gray to whitepart to part4 : 3
white to graypart to part3 : 4
gray to all pebblespart to whole4 : 7
white to all pebblespart to whole3 : 7

Another handful adds another column just like the others. So however many handfuls Felix collects, the gray pebbles are 4 out of every 7, which is \displaystyle \frac{4}{7} of the bucket.

A part sits inside the whole, so in a part-to-whole ratio the first number can never be bigger than the second. Two parts sit side by side, so in a part-to-part ratio either may be bigger: 4 : 3 and 3 : 4 are both correct, read in opposite orders.

Results

From the parts to the whole. When a collection has exactly two parts in the ratio 2 : 5, each part compares with the whole as 2 : 7 and 5 : 7, because 2 + 5 = 7.

From the whole to the other part. When one of exactly two parts compares with the whole as 3 : 7, the other part compares with the whole as 4 : 7, because 7 - 3 = 4, and the two parts compare as 3 : 4.

A part-to-whole ratio is a fraction. A part that compares with the whole as 2 : 7 is \displaystyle \frac{2}{7} of the whole. The fractions of all the parts add to 1: \displaystyle \frac{2}{7} + \frac{5}{7} = 1.

Three or more parts. When the parts are in the ratio 2 : 3 : 4, each pair keeps its own numbers (2 : 3, 3 : 4 and 2 : 4), and each part compares with the whole using the sum 9: as fractions, \displaystyle \frac{2}{9}, \displaystyle \frac{3}{9} and \displaystyle \frac{4}{9}.

Why it is true

“For every 2 red there are 5 white” means the bunting can be cut into repeats that are all alike, each holding 2 red flags and 5 white ones, 7 flags in all.

The repeats are all the same size, so they are equal shares of the string. Inside each share, 2 of the 7 flags are red. Put the shares back together and red is still 2 out of every 7, so red is \displaystyle \frac{2}{7} of the string, whatever the number of repeats. In the same way white is \displaystyle \frac{5}{7}, and since a repeat holds nothing but red and white flags, \displaystyle \frac{2}{7} + \frac{5}{7} is the whole, 1.

Why is \displaystyle \frac{2}{5} not the red part? A fraction of the string needs the size of the whole string underneath, and 5 counts only the white flags; the red flags are not among them. What \displaystyle \frac{2}{5} does measure is red against white: in each repeat, 2 is \displaystyle \frac{2}{5} of 5.

Three parts work the same way. A ratio of 2 : 3 : 4 cuts the collection into alike groups of 9, so the first part is \displaystyle \frac{2}{9} of the whole. Cover up the third part in every group and what remains holds 2 of the first for every 3 of the second, which is why that pair keeps the ratio 2 : 3.

Worked examples

Example 1

Four ratios from one string of bunting

Marisol’s bunting in Figure 2 repeats 2 red flags and 5 white flags. Write the ratio of red to white, white to red, red to all flags and white to all flags. What fraction of the flags is red, and what fraction is white?

2 red + 5 white = 7 flags 2 red + 5 white = 7 flags
Figure 2Marisol’s bunting. Every repeat of the pattern holds 2 red flags and 5 white ones, 7 flags in all.
Solution

Red and white are both parts, so comparing them gives part-to-part ratios, in the order asked:

\text{red} : \text{white} = 2 : 5 \qquad \text{white} : \text{red} = 5 : 2

Each repeat holds 2 + 5 = 7 flags, so the whole’s number is 7, and the part-to-whole ratios are red to all 2 : 7 and white to all 5 : 7. As fractions, \displaystyle \frac{2}{7} of the flags are red and \displaystyle \frac{5}{7} are white.

Check. The two fractions cover the whole string: \displaystyle \frac{2}{7} + \frac{5}{7} = 1.

Example 2

Starting from the whole

Ayumi sorts the pebbles from a beach into gray and white, and nothing else. Out of every 7 pebbles, 3 are white. Write the ratio of white to gray, gray to white and gray to all pebbles, and the fraction of the pebbles that are gray.

Solution

“3 out of every 7” compares the white part with the whole: white to all is 3 : 7. It is not a part-to-part ratio, because the 7 includes the white pebbles themselves.

There are only two parts, so out of every 7 pebbles, 7 - 3 = 4 are gray. Now both parts are known:

\text{white} : \text{gray} = 3 : 4 \qquad \text{gray} : \text{white} = 4 : 3

Gray to all pebbles is 4 : 7, so \displaystyle \frac{4}{7} of the pebbles are gray.

Example 3

Counting the parts from the whole

Marisol’s finished string has 35 flags, made of whole repeats of 2 red and 5 white. How many flags are red and how many are white?

Solution

Each repeat holds 7 flags, so the number of repeats is 35 \div 7 = 5. Each repeat gives 2 red flags and 5 white ones:

\text{red} = 5 \times 2 = 10 \qquad \text{white} = 5 \times 5 = 25

The part-to-whole ratio gives the same answer in one line: red is \displaystyle \frac{2}{7} of the string, and \displaystyle \frac{2}{7} of 35 is 10.

Check. 10 + 25 = 35. Drawing the repeats as equal blocks is the method of Tape diagrams and sharing in a ratio.

Example 4

Testing a claim about two fifths

A party leaflet says: “The red and white flags are in the ratio 2 : 5, so \displaystyle \frac{2}{5} of the flags are red.” Test the claim on Marisol’s 35-flag string, and say what \displaystyle \frac{2}{5} does describe.

Solution

If \displaystyle \frac{2}{5} of the flags were red, the string would have \displaystyle \frac{2}{5} of 35, which is 14, red flags. The last example found 10, so the claim fails. The real red part is

\frac{10}{35} = \frac{2}{7}

which matches red to all, 2 : 7. The \displaystyle \frac{2}{5} compares red with white: \displaystyle \frac{2}{5} of the 25 white flags is 10, the number of red flags. So 2 : 5 says there are \displaystyle \frac{2}{5} as many red flags as white flags, and nothing about \displaystyle \frac{2}{5} of the string.

Example 5

A vote with three choices

Zainab’s class of 27 votes on a trip. The votes for the museum, the farm and the science center are in the ratio 2 : 3 : 4, and every student votes once. How many votes did each place get? Write the ratio of farm to science center votes, museum votes to all votes, and museum votes to the votes that were not for the museum.

Museum Farm Science center 9 votes 9 votes 9 votes
Figure 3The class vote in groups of 9. Each group holds 2 museum votes, 3 farm votes and 4 science center votes.
Solution

The votes fall into alike groups of 2 museum, 3 farm and 4 science center votes, 9 votes per group (Figure 3). There are 27 \div 9 = 3 groups:

\begin{aligned} \text{museum} &= 3 \times 2 = 6 \\ \text{farm} &= 3 \times 3 = 9 \\ \text{science center} &= 3 \times 4 = 12 \end{aligned}

and 6 + 9 + 12 = 27, as it must.

Farm to science center keeps its two numbers: 3 : 4. Museum to all votes is part to whole, 2 : 9, so \displaystyle \frac{2}{9} of the class chose the museum; the counts agree, since \displaystyle \frac{6}{27} = \frac{2}{9}.

The votes not for the museum are the farm and science center votes together, 3 + 4 = 7 in each group. Museum to not museum is 2 : 7. This is part to part, even though the second part joins two parts: the museum votes are not among the 7.

Example 6

From fractions to a three-part ratio

Pieter sorts pebbles from the beach. One quarter are white, one third are speckled and the rest are gray. Find the ratio of white to speckled to gray pebbles.

Solution

Fractions of the whole are part-to-whole ratios, so put them over one whole. Twelfths can show both quarters and thirds:

\frac{1}{4} = \frac{3}{12} \qquad \frac{1}{3} = \frac{4}{12}

The gray pebbles are the rest of the whole:

\frac{12}{12} - \frac{3}{12} - \frac{4}{12} = \frac{5}{12}

So out of every 12 pebbles, 3 are white, 4 speckled and 5 gray: the ratio of white to speckled to gray is 3 : 4 : 5.

Check. 3 + 4 + 5 = 12, and white to all is 3 : 12, which is \displaystyle \frac{1}{4} as given.

Boundaries

The parts must cover the whole and must not overlap. Adding the parts to get the whole works only when every item is in exactly one part. If the museum and farm votes are 2 : 3 but some students chose the science center, 2 + 3 = 5 is not the class, and the museum’s share of the class cannot be found. If a pebble can be both gray and speckled, adding the gray and speckled counts counts it twice.

Measured amounts must share a unit. 2 liters of red paint to 5 liters of white is \displaystyle \frac{2}{7} red, like the bunting. With 2 liters of red and 5 milliliters of white, 2 + 5 means nothing until both are in the same unit.

Whole repeats give only some totals. Whole repeats of 2 red and 5 white make 7, 14, 21 flags and so on. A string of 30 flags cannot be made that way.

Where this goes next

Splitting a total in a ratio, as in the bunting and vote examples, is the main use of Tape diagrams and sharing in a ratio. Why 10 red to 25 white describes the same bunting as 2 : 5 is the subject of Scaling a ratio up or down. A percent is a part-to-whole ratio whose whole is always 100 (What percent means: a rate per hundred). To say which of two strings is redder, compare their part-to-whole fractions (Deciding which of two ratios is larger). In probability the same split returns: odds compare outcomes part to part, and a probability is the part-to-whole fraction.

Exercises

Two parts

  1. Felix hangs bunting with 3 blue flags for every 4 yellow flags, and no other colors. Write the ratio of blue to yellow, yellow to blue, blue to all flags, and yellow to all flags.
  2. Ayumi’s bucket holds 8 shells and 5 pebbles and nothing else. Write the ratio of shells to pebbles, pebbles to everything in the bucket, and shells to everything in the bucket.
  3. Say whether each statement gives a part-to-part or a part-to-whole ratio. (a) 5 out of every 9 flags are green. (b) For every 2 votes for the museum there were 7 votes for somewhere else. (c) The ratio of gray pebbles to all pebbles is 4 : 11. (d) There are 3 white pebbles for every 1 gray pebble.
  4. In a bucket, 4 out of every 9 pebbles are speckled and the rest are plain. Write the ratio of speckled to plain pebbles and of plain pebbles to all pebbles.
  5. In a class vote every student voted yes or no, and 11 out of every 20 voted yes. Write the ratio of yes votes to no votes.

Ratios and fractions

  1. Bunting has red and white flags only, in the ratio 3 : 5. What fraction of the flags is red, and what fraction is white?
  2. Two ninths of the pebbles in a bucket are white and the rest are gray. Write the ratio of white to gray and of gray to all pebbles.
  3. A bucket holds only gray and white pebbles, gray to white 4 : 3. What fraction of the pebbles is gray? Explain why \displaystyle \frac{4}{3} cannot be the answer.
  4. A vote had two choices, and the museum and farm votes were in the ratio 2 : 3. What fraction of the votes went to the farm, and what fraction to the museum?
  5. A string of 42 flags is made of whole repeats of 2 red and 5 white. How many flags of each color are there?
  6. Pieter’s bunting has 40 flags. Three eighths of them are blue and the rest are white. Write the ratio of blue to white, and find how many flags of each color there are.

Three or more quantities

  1. Votes for the museum, the farm and the science center are in the ratio 2 : 3 : 4. Write the ratio of farm to museum votes and of science center votes to all votes, and the fraction of the votes that went to the museum.
  2. A class of 36 votes for the museum, the farm and the science center in the ratio 2 : 3 : 4. How many votes does each get?
  3. A string of 30 flags is made of whole repeats of red, white and blue flags in the ratio 1 : 2 : 3. How many flags of each color are there, and what fraction is white?
  4. Gray, white and speckled pebbles are in the ratio 5 : 2 : 1. What fraction of the pebbles is speckled? If 40 pebbles are collected in whole handfuls of that make-up, how many of each are there?
  5. Half of a bucket of pebbles is gray, one third is white and the rest is speckled. Find the ratio of gray to white to speckled.
  6. Callum’s bunting has only red, white and blue flags. Red to white is 2 : 5 and white to blue is 5 : 3, counting the same 5 white flags in each repeat. Write the ratio of red to white to blue, and the fraction of the flags that is blue.

Working back to the whole

  1. Felix’s handfuls always hold 4 gray and 3 white pebbles. Can a bucket of whole handfuls hold exactly 30 pebbles? Explain.
  2. Votes for the museum, the farm and the science center are in the ratio 2 : 3 : 4, and the science center got 6 more votes than the museum. How many students voted?

Explaining

  1. Explain why the first number of a part-to-whole ratio can never be bigger than the second, while either number of a part-to-part ratio can be the bigger one.
  2. The museum and farm votes are in the ratio 2 : 3, but some students voted for the science center. Can you tell what fraction of the class voted for the museum? Explain, and say what \displaystyle \frac{2}{5} does describe.
  3. In one string red to white is 1 : 1. In another, red to all flags is 1 : 2. Do the two strings have the same fraction of red flags? Explain.
Answers
  1. 3 : 4, 4 : 3, 3 : 7 and 4 : 7. Each repeat has 3 + 4 = 7 flags.
  2. 8 : 5, 5 : 13 and 8 : 13, since 8 + 5 = 13.
  3. (a) Part to whole: green against all flags. (b) Part to part: the other votes do not include the museum votes; the whole is 9. (c) Part to whole. (d) Part to part: white against gray.
  4. 4 : 5 and 5 : 9. Out of every 9, 9 - 4 = 5 are plain.
  5. 11 : 9. Out of every 20 votes, 20 - 11 = 9 were no.
  6. Red \displaystyle \frac{3}{8} and white \displaystyle \frac{5}{8}. Each repeat has 3 + 5 = 8 flags.
  7. 2 : 7 and 7 : 9. Out of every 9, 9 - 2 = 7 are gray.
  8. \displaystyle \frac{4}{7}, since each handful of 4 + 3 = 7 has 4 gray. \displaystyle \frac{4}{3} is more than 1, which would mean more gray pebbles than pebbles; it compares gray with white only.
  9. Farm \displaystyle \frac{3}{5} and museum \displaystyle \frac{2}{5}. With only two choices the whole is 2 + 3 = 5, so here \displaystyle \frac{2}{5} is the museum’s fraction.
  10. 12 red and 30 white. There are 42 \div 7 = 6 repeats: 6 \times 2 = 12 and 6 \times 5 = 30.
  11. 3 : 5, since out of every 8 flags 3 are blue and 5 white. Blue is \displaystyle \frac{3}{8} of 40, which is 15, and white is 40 - 15 = 25.
  12. 3 : 2, 4 : 9 and \displaystyle \frac{2}{9}. Each group holds 2 + 3 + 4 = 9 votes.
  13. Museum 8, farm 12, science center 16. There are 36 \div 9 = 4 groups of 9.
  14. 5 red, 10 white and 15 blue; white is \displaystyle \frac{1}{3}. Each repeat has 6 flags, so there are 30 \div 6 = 5 repeats, and \displaystyle \frac{10}{30} = \frac{1}{3}.
  15. \displaystyle \frac{1}{8}, since 5 + 2 + 1 = 8. With 40 \div 8 = 5 handfuls there are 25 gray, 10 white and 5 speckled.
  16. 3 : 2 : 1. In sixths, gray is \displaystyle \frac{3}{6}, white \displaystyle \frac{2}{6} and speckled \displaystyle \frac{6}{6} - \frac{3}{6} - \frac{2}{6} = \frac{1}{6}.
  17. 2 : 5 : 3, and blue is \displaystyle \frac{3}{10}. Each repeat has 2 red, 5 white and 3 blue, 10 flags.
  18. No. Whole handfuls give 7, 14, 21, 28, 35 and so on. Four handfuls make 28 and five make 35, so 30 is never reached.
  19. 27 students. In each group of 9 votes the science center has 4 - 2 = 2 more than the museum, so 6 more votes means 6 \div 2 = 3 groups, and 3 \times 9 = 27.
  20. The part is counted inside the whole, so the whole has at least as many items as the part. Two parts are separate groups, so either can be larger, and the same two parts give 4 : 3 or 3 : 4 depending on the order they are named.
  21. No. The class is the museum, farm and science center voters together, and 2 : 3 says nothing about the science center, so the whole’s number is unknown. The \displaystyle \frac{2}{5} is the museum’s fraction of the students who chose the museum or the farm.
  22. Yes, both are half red. Red to white 1 : 1 gives repeats of 1 + 1 = 2 flags with 1 red, so red is \displaystyle \frac{1}{2}. Red to all 1 : 2 is already the fraction \displaystyle \frac{1}{2}.
kg1128 · updated 2026-10-12 · CC-BY-SA-4.0