CDS Mathematics · Ratio, Proportion and Variation

Combining and Dividing by Ratios

Chain ratios through a common term, and divide a quantity in a ratio by finding the value of one part.

All Ratio, Proportion and Variation notes CDS Mathematics strategy

Why this matters

Sixteen PYQs, almost all MODERATE or EASY. Two skills do the work: scaling ratios so the shared term matches (A : B and B : C into A : B : C), and splitting a total once you know how many parts it has. Wills and inheritance chains are the same skill written as a story.

Concept 1 of 2: Chaining ratios

A : B

and

B : C

can only be joined when

B

has the same number in both. Scale each ratio so the shared term matches, then read off

A : B : C

Definition

To join $A : B = p : q$ and $B : C = r : s$ , write $A : B : C = pr : qr : qs$ .
Chains of fractions multiply: $\dfrac ad = \dfrac ab\cdot\dfrac bc\cdot\dfrac cd$ .
Pairwise sums in a ratio: if $(a + b) : (b + c) : (c + a) = p : q : r$ , all the parts together stand for $2(a + b + c)$ .
Fractions of fractions ('half of one-third'): multiply them.

Joining two ratios

A : B = p : q,\ B : C = r : s \;\Rightarrow\; A : B : C = pr : qr : qs

Worked example

A : B = 2 : 3

and

B : C = 4 : 7

, find

A : B : C

and

A : C

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2018 · CDS (I) 2018 — Elementary Mathematics · Q24Easy

Example 1 · Ratio, Proportion and Variation · Ratio and Proportion

A : B = 1 : 2

B : C = 3 : 4

C : D = 2 : 3

and

D : E = 3 : 4

, then what is

B : E

equal to?

Scale, do not add

Joining

2 : 3

and

4 : 5

2 : 7 : 5

is wrong. Multiply each ratio so the shared term is the same number:

8 : 12 : 15

Drill 6 more on chaining ratios

Concept 2 of 2: Dividing a quantity in a ratio

Dividing in the ratio

p : q : r

cuts the total into

p + q + r

equal parts. Find one part, then multiply. When shares are described as multiples of one another, name the smallest share and add everything up.

Definition

The share of $p$ when $T$ is split $p : q : r$ is $\dfrac{p}{p + q + r}\,T$ .
Ratios given as fractions $\tfrac12 : \tfrac13 : \tfrac14$ : multiply by the LCM of the denominators first ( $6 : 4 : 3$ ).
Two numbers in the ratio $m : n$ are $mk$ and $nk$ ; a product or sum of squares then fixes $k$ .
'Each son gets twice a daughter, each daughter as much as the mother': call the mother's share $w$ and count.

Share in a ratio

\text{share} = \dfrac{p}{p + q + r}\times T

Worked example

Rs. 6,300 is left to a widow, three sons and two daughters. Each son gets twice a daughter, and each daughter gets as much as the widow. Find a son's share.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2021 · CDS (II) 2021 — Elementary Mathematics · Q26Easy

Example 2 · Ratio, Proportion and Variation · Ratio and Proportion

If three positive numbers are in the ratio

2:3:5

and the sum of their squares is 1368, then what is sum of the numbers ?

Answer the share that was asked

When amounts are deducted before a ratio applies, the share asked for is the amount BEFORE the deduction. Solving for the reduced amount and stopping there gives a number that looks right but is not the share.

Drill 8 more on dividing a quantity in a ratio

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Chaining ratios
Joining two ratios
$A : B = p : q,\ B : C = r : s \;\Rightarrow\; A : B : C = pr : qr : qs$
Dividing a quantity in a ratio
Share in a ratio
$\text{share} = \dfrac{p}{p + q + r}\times T$

Watch out for (2)

Scale, do not add
→ Chaining ratios
Answer the share that was asked
→ Dividing a quantity in a ratio

Test yourself on Ratio, Proportion and Variation

20 past CDS questions from this chapter, timed at 24 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Ratio, Proportion and Variation chapter test Drill all 16 questions