Technique

Introduce a common factor $k$ for a ratio

Arithmetic • Ratio and Proportion

Formula

a : b : c = p : q : r \Rightarrow a = p k, b = q k, c = r k

If

a : b : c = p : q : r

, then

a = p k

,

b = q k

,

c = r k

for some

k > 0

. A ratio fixes the quantities only up to this common factor.

How to use it

1Use when a ratio is given but no absolute value is.
2In homogeneous expressions $k$ cancels, so it never needs to be found.
3If one absolute value is also given, solve for $k$ first.

Worked example

If

x : y = 3 : 5

and

x + y = 32

, find

y

.

Step-by-step

1
$x = 3 k, y = 5 k$
Why: A ratio fixes $x$ and $y$ only up to a common factor.
2
$3 k + 5 k = 32 \Rightarrow 8 k = 32 \Rightarrow k = 4$
3
$y = 5 k = 20$

Answer

20

Watch out

Only one

k

per ratio chain. Two independent ratios need two letters unless a common term links them.

Also known as

ratio to variables, common multiplier, let the terms be 3k and 5k

Appears in past papers

2020 (II)

Q25 Q37 Q71 Q77 Q78

2020 (I)

Q17 Q28 Q30 Q33 Q34 Q36 Q44 Q68 Q76

2019 (II)

Q22 Q23 Q46 Q47 Q49

2019 (I)

Q27 Q32 Q43 Q45 Q71 Q80 Q82

2018 (II)

Q20 Q30 Q31 Q32 Q36 Q37 Q47 Q54 Q56 Q86

2018 (I)

Q24 Q37 Q42 Q50 Q68 Q87 Q97

2017 (II)

Q5 Q41 Q42 Q56 Q73

2017 (I)

Q12 Q22 Q32 Q39 Q49 Q50 Q72

2016 (II)

Q12 Q16 Q35 Q36 Q43 Q50 Q51 Q86