Depth · one concept from first principles · step 2 of 4

Chain two ratios, then evaluate by ratios

If $a:b=2:1$ and $b:c=2:3$ , find the value of $\frac{a(ab+bc)+b(bc+ca)+c(ca+ab)}{abc}$ .

RatiosExpressions

Your answer

How to solve

Strategy Organize Information in More Ways — Only ratios are known, so rewrite the expression in terms of ratios. Splitting the big fraction term by term turns it into three ratios plus 3; chaining the two given ratios supplies all three.

1STEP 1

Chain the two ratios

Make b the same in both: a:b=2:1=4:2 and b:c=2:3.

a:b:c=4:2:3

2STEP 2

Split the fraction

Expand the top and divide each piece by abc.

(a²b+abc)/(abc)+(b²c+abc)/(abc)+(c²a+abc)/(abc)=a/c+b/a+c/b+3

3STEP 3

Read off the ratios

From a:b:c=4:2:3.

a/c=4/3, b/a=1/2, c/b=3/2

4STEP 4

Add

1/2+3/2=2, and 2+3=5.

4/3+1/2+3/2+3=19/3

Answer

19/3

Use actual numbers in the ratio, a=4, b=2, c=3: the top is 4 × 14+2 × 18+3 × 20=152 and abc=24, and 152/24=19/3.

Takeaway

If only ratios are given, rewrite the expression in ratios and the unknown size disappears.

Chain the two ratios
Split the fraction
Read off the ratios
Add

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