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Ratio, proportion & rates of change · Ratio

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Combining and relating ratios

How to link two ratios that share a quantity into one three-part ratio, and how ratios connect to fractions and to the "1 : n" form.

Ratio, proportion & rates of change · Ratio

Combining and relating ratios

How to link two ratios that share a quantity into one three-part ratio, and how ratios connect to fractions and to the "1 : n" form.

Why it works

Often two ratios share a common quantity: "a:b=3:4a : b = 3 : 4 and b:c=6:5b : c = 6 : 5". Both mention bb, but they disagree about its size — one calls it 44, the other 66. You cannot just staple them into 3:4:53 : 4 : 5, because in a valid three-part ratio the middle number has to mean the same amount of bb in both halves.

The fix is to scale each ratio so the shared number matches. Make bb the same in both by taking a common multiple of 44 and 66, which is 1212:

a:b=3:4  →×3  9:12,b:c=6:5  →×2  12:10.a : b = 3 : 4 \;\xrightarrow{\times 3}\; 9 : 12, \qquad b : c = 6 : 5 \;\xrightarrow{\times 2}\; 12 : 10.

Now both say b=12b = 12, so they lock together: a:b:c=9:12:10a : b : c = 9 : 12 : 10. (You can divide through by a common factor at the end if there is one; here there isn't.) This is the same idea as finding a common denominator before adding fractions — line up the shared piece, then read across.

Chains of ratios let you find real amounts. "Managers : staff =1:8= 1 : 8, staff : trainees =4:3= 4 : 3, and there are 9696 staff." You don't even need the combined ratio — work outward from the known quantity. Managers =968=12= \frac{96}{8} = 12; trainees =96×34=72= 96 \times \frac{3}{4} = 72. Each ratio is a bridge from a known amount to an unknown one.

Ratios and fractions are two views of the same split. A ratio a:ba : b says the whole is a+ba + b parts, so the first quantity is aa+b\frac{a}{a+b} of the total and the second is ba+b\frac{b}{a+b}. Walkers to non-walkers 3:73 : 7 means walkers are 310\frac{3}{10} of the class — the denominator is the total 3+7=103 + 7 = 10, a favourite place to go wrong (it is not 37\frac{3}{7}).

The "1:n1 : n" form is for comparing and for scale. To write a ratio as 1:n1 : n, divide both numbers by the first. A map where 44 cm represents 1010 km is, in consistent units, 4 cm:1 000 000 cm4\text{ cm} : 1\,000\,000\text{ cm}; dividing both by 44 gives 1:250 0001 : 250\,000. Converting to the same unit before dividing is the step people skip.

Ratios can hold algebra. "(2x+1):(x+4)=3:2(2x + 1) : (x + 4) = 3 : 2" is just two equal ratios, so cross-multiply: 2(2x+1)=3(x+4)2(2x + 1) = 3(x + 4), giving x=10x = 10. The ratio is only a wrapper around an ordinary equation.