C1.11; E1.11 - Simplifying and sharing ratios

C1.11; E1.11 - Simplifying and sharing ratios

A ratio compares quantities by equal-sized parts. You will put quantities into common units, simplify ratios with two or three terms, and divide a total fairly according to a stated ratio.

Ratios and common units

A mix uses 1.2 kg1.2\,\mathrm{kg} of flour and 300 g300\,\mathrm{g} of sugar. Is there really less flour because 1.21.2 is smaller than 300300? The units tell a different story. You must compare the same kind of quantity in the same units.

Ratio

A ratio compares the relative sizes of quantities in a stated order. Flour : sugar =4:1=4:1 means four equal parts of flour for each one equal part of sugar. The parts have the same size, but that size is not fixed by the ratio alone.

Order matters: flour : sugar =4:1=4:1 is the same comparison as sugar : flour =1:4=1:4. It is not the same recipe as flour : sugar =1:4=1:4.

Worked example 1: kilograms and grams

Question. Write the ratio of 1.2 kg1.2\,\mathrm{kg} of flour to 300 g300\,\mathrm{g} of sugar in its simplest form.

Plan. Convert kilograms to grams, then divide both terms by the same number.

1.2 kg=1200 guse 1 kg=1000 gflour : sugar=1200:300compare common units=4:1divide both terms by 300\begin{aligned} 1.2\,\mathrm{kg}&=1200\,\mathrm{g} &&\text{use }1\,\mathrm{kg}=1000\,\mathrm{g}\\ \text{flour : sugar}&=1200:300 &&\text{compare common units}\\ &=4:1 &&\text{divide both terms by 300} \end{aligned}

Answer. 4:14:1.

Check. 4×300 g=1200 g=1.2 kg4\times300\,\mathrm{g}=1200\,\mathrm{g}=1.2\,\mathrm{kg}. There is four times as much flour as sugar.

Now increase only the sugar mass by a factor of ten. That changes the ratio, even though the unit conversion stays the same.

Worked example 2: the same units, a different comparison

Question. Write the ratio of 1.2 kg1.2\,\mathrm{kg} of flour to 3000 g3000\,\mathrm{g} of sugar in its simplest form.

Plan. Compare both masses in grams, then find a common factor.

flour : sugar=1200:3000convert the flour to grams=2:5divide both terms by 600\begin{aligned} \text{flour : sugar}&=1200:3000 &&\text{convert the flour to grams}\\ &=2:5 &&\text{divide both terms by 600} \end{aligned}

Answer. 2:52:5.

Check. Two parts of 600 g600\,\mathrm{g} give 1200 g1200\,\mathrm{g}; five give 3000 g3000\,\mathrm{g}. Flour is now the smaller quantity, as the answer shows.

Watch out

1.2:300=1:2501.2:300=1:250 does not compare these masses correctly: the first number counts kilograms and the second counts grams. Convert first. You may use either common unit, provided both quantities use it.

Simplifying ratios

Equivalent ratios describe the same comparison using differently sized parts. For example, 18:3018:30 and 3:53:5 have the same meaning because every term has been divided by 66. Dividing only one term would change the comparison.

Simplest form of a ratio

A ratio is in simplest form when its terms are whole numbers with no common factor greater than 11. For a three-term ratio, consider factors common to all three terms.

Decimal terms can be cleared by multiplying every term by 1010, 100100, or another suitable power of ten. Fractional terms can be cleared by multiplying every term by a common denominator. For instance, 12:34\frac12:\frac34 becomes 2:32:3 after multiplying both terms by 44. Then simplify the whole-number ratio if needed.

Worked example 3: removing decimals

Question. Simplify 0.6:0.90.6:0.9.

Plan. Make both terms whole numbers before removing their common factor.

0.6:0.9=6:9multiply both terms by 10=2:3divide both terms by 3\begin{aligned} 0.6:0.9&=6:9 &&\text{multiply both terms by 10}\\ &=2:3 &&\text{divide both terms by 3} \end{aligned}

Answer. 2:32:3.

Check. Multiplying both answer terms by 0.30.3 gives 0.6:0.90.6:0.9, recovering the original comparison.

Add a third quantity while retaining the first two. Every operation must now affect all three terms.

Worked example 4: three quantities

Question. Simplify 0.6:0.9:1.50.6:0.9:1.5.

Plan. Clear the decimals in all three terms and divide by their highest common factor.

0.6:0.9:1.5=6:9:15multiply every term by 10=2:3:5divide every term by 3\begin{aligned} 0.6:0.9:1.5&=6:9:15 &&\text{multiply every term by 10}\\ &=2:3:5 &&\text{divide every term by 3} \end{aligned}

Answer. 2:3:52:3:5.

Check. Multiplying all three terms by 0.30.3 recovers 0.6:0.9:1.50.6:0.9:1.5. No integer greater than 11 divides all of 2,3,52,3,5.

Watch out

Subtracting a fixed number does not preserve a ratio: 6:96:9 does not simplify to 3:63:6. The first comparison has value 6/9=2/36/9=2/3, while the second has value 3/6=1/23/6=1/2. Use multiplication or division by the same non-zero factor.

Sharing a quantity in a ratio

When a total is shared in a ratio, add the ratio terms to find how many equal parts make the whole total. Divide the total by that number, then multiply by the number of parts each person receives.

For 2:52:5, the first share is 2/72/7 of the total, not 2/52/5: the total consists of both shares together. This is the link between ratios and fractions.

[DIAGRAM: asset_name: Seven equal ratio parts; asset_slug: ratio_equal_parts; caption: A 2 to 5 share contains seven equal parts altogether.; description: Seven equal adjacent rectangles form one bar. The first two are pale blue and labelled first share, 2 parts. The last five are pale green and labelled second share, 5 parts. A bracket beneath the whole bar is labelled total, 7 equal parts.]
A 2 to 5 share contains seven equal parts altogether.

Worked example 5: two shares

Question. Divide \84betweenAnaandBointheratioAna:Bobetween Ana and Bo in the ratio Ana : Bo=2:5$.

Plan. Find the value of one of the seven equal parts.

2+5=7total number of parts84÷7=12dollars per part2×12=24Ana receives two parts5×12=60Bo receives five parts\begin{aligned} 2+5&=7 &&\text{total number of parts}\\ 84\div7&=12 &&\text{dollars per part}\\ 2\times12&=24 &&\text{Ana receives two parts}\\ 5\times12&=60 &&\text{Bo receives five parts} \end{aligned}

Answer. Ana receives \24;Boreceives; Bo receives $60$.

Check. 24+60=8424+60=84, and 24:60=2:524:60=2:5. Check both the total and the ratio.

Keep the total and the first two ratio terms, but introduce a third recipient. The total number of parts changes, so the old part value cannot be reused.

Worked example 6: three shares

Question. Divide \84betweenAna,BoandCyintheratiobetween Ana, Bo and Cy in the ratio2:5:7$.

Plan. Share the total among fourteen equal parts.

2+5+7=14all three shares make the total84÷14=6dollars per part2×6=12Ana’s share5×6=30Bo’s share7×6=42Cy’s share\begin{aligned} 2+5+7&=14 &&\text{all three shares make the total}\\ 84\div14&=6 &&\text{dollars per part}\\ 2\times6&=12 &&\text{Ana's share}\\ 5\times6&=30 &&\text{Bo's share}\\ 7\times6&=42 &&\text{Cy's share} \end{aligned}

Answer. Ana receives \12,Bo, Bo $30,andCy, and Cy $42$.

Check. 12+30+42=8412+30+42=84, and dividing the shares by 66 gives 2:5:72:5:7.

The value of one part need not be a whole number. Keep it exact until the shares are found. If a problem involves indivisible objects, such as people, check that the proposed shares are whole numbers before accepting the model.

Watch out

The number you divide by is the sum of the ratio terms. Dividing 8484 by 55 in the first example treats the second share as if it were the whole total.

Quick Recap

  • Compare like quantities in common units and preserve the stated order.
  • Multiply or divide every ratio term by the same non-zero factor.
  • Simplest form has whole-number terms with no common factor greater than 11.
  • To share a total, find the total number of equal parts, then the value of one part.

The opening mixture contains four times as much flour as sugar: 1.2 kg:300 g=1200:300=4:11.2\,\mathrm{kg}:300\,\mathrm{g}=1200:300=4:1.