Dividing a Total in a Ratio, and… — AFCAT Numerical Ability Notes

Concept 1 of 5: Dividing a total in a ratio

Dividing ₹1,200 in the ratio

2 : 3 : 5

cuts it into

2 + 3 + 5 = 10

equal parts. One part is ₹120, and each person gets their number of parts.

Definition

The share for the term $p$ when $T$ is split $p : q : r$ is $\dfrac{p}{p + q + r} \times T$ .
Given one share instead of the total: that share divided by its term is one part. Multiply up from there.
The difference between two shares is the difference of their terms times one part.
'A gets $\tfrac 29$ of what B and C together get' is the ratio $A : (B + C) = 2 : 9$ , so A has 2 of 11 parts.
Money, weight and people all divide the same way. If the answer is asked in another unit, convert at the end.

Share in a ratio

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

Worked example

₹1,200 is divided among X, Y and Z in the ratio

2 : 3 : 5

. Find each share, and how much more Z gets than X.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

AFCAT · 2018 · AFCAT (I) 2018 — Numerical Ability · 04-304Easy

Example 1 · Ratio and Proportion · Dividing in a Ratio and Partnership

A box filled with paper bundles weighs 36 kg. If the weights of the box and the paper bundles are in the ratio of 3 : 22, then the weight of the papers (in g) is

Answer in the unit asked

A 20 kg load split

1 : 4

has 16 kg in the larger part, which is 16,000 g. If the options are in grams, 16 is not among them.

A share of 'the other two together'

'A gets

\tfrac 29

of what B and C get together' makes A 2 parts out of 11, not 2 out of 9.

Drill 10 more on dividing a total in a ratio

Concept 2 of 5: Dividing in a ratio of fractions

A ratio such as

\tfrac 13 : \tfrac 14 : \tfrac 16

is still a ratio. Clear the fractions first, by multiplying by the LCM of the denominators, then divide as usual. 'A third of the first equals a quarter of the second' says the same thing the other way round.

Definition

$\tfrac 1a : \tfrac 1b : \tfrac 1c$ : multiply every term by the LCM of $a, b, c$ . For example $\tfrac 13 : \tfrac 14 : \tfrac 16 = 4 : 3 : 2$ .
'A third of A equals a quarter of B' means $\tfrac A3 = \tfrac B4 = k$ , so $A = 3k$ , $B = 4k$ and $A : B = 3 : 4$ .
'Three times the first equals four times the second' is the equal-multiples case from Combining Ratios.

Clearing fractions

\tfrac 1a : \tfrac 1b : \tfrac 1c = \tfrac La : \tfrac Lb : \tfrac Lc \quad (L = \text{LCM of } a, b, c)

Worked example

Divide ₹1,350 among three people in the ratio

\tfrac 13 : \tfrac 14 : \tfrac 16

.

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The same idea in a practice question:

Practice questionEasy

Example 2 · Ratio and Proportion · Dividing in a Ratio and Partnership

Divide 1870 into three parts in such a way that half of the first part, one-third of the second part and one-sixth of the third part are equal. Then, the third part is

The denominators are not the ratio

In

\tfrac 12 : \tfrac 13

the first person gets more, 3 parts to 2. Reading the denominators as

2 : 3

turns the ratio round.

Drill 3 more on dividing in a ratio of fractions

Concept 3 of 5: Numbers in a ratio with one more condition

Write the numbers as

2k, 3k, 4k

and turn the condition into one equation in

k

. A sum gives

k

directly, a sum of squares gives

k^2

, and a product gives

k^3

.

Definition

For numbers $pk, qk, rk$ : the sum is $(p + q + r)k$ , the sum of squares $(p^2 + q^2 + r^2)k^2$ , and the product $pqr\,k^3$ .
Shares described by differences ('A gets ₹7 more than B'): name the smallest share and build the others from it.
'Boys exceed girls by 20% of the total' is an equation: $B - G = 0.2(B + G)$ . Solve it for $B : G$ (here $3 : 2$ ).

Numbers from a ratio

x : y : z = p : q : r \;\Rightarrow\; x = pk,\ y = qk,\ z = rk

Worked example

Three numbers are in the ratio

1 : 3 : 4

and the sum of their squares is 416. Find them.

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The same idea in a practice question:

Practice questionModerate

Example 3 · Ratio and Proportion · Dividing in a Ratio and Partnership

In a class, the number of boys is more than the number of girls by 12% of the total students. What is the ratio of number of boys to that of girls?

A sum of squares gives k squared

If

2k, 3k, 4k

have squares adding to 116, then

29k^2 = 116

, so

k^2 = 4

and

k = 2

. Stopping at

k = 4

doubles every number.

Drill 4 more on numbers in a ratio with one more condition

Concept 4 of 5: A changed ratio, or shares with an amount taken off

Two ways of splitting the same total are compared share by share. Work out each person's share both ways; the gain or loss is the difference.

Definition

Same total, two ratios: the share under one minus the share under the other.
If one person's loss is given, write it as $\left(\dfrac{q}{p + q} - \dfrac{s}{r + s}\right)T$ and solve for $T$ .
'If each had received ₹10 less, the shares would be $2 : 3 : 4$ ': take ₹10 from each first, split what is left, then add the ₹10 back.

Gain or loss from a changed ratio

\text{change} = \left(\dfrac{q}{p + q} - \dfrac{s}{r + s}\right) T

Worked example

₹2,100 was to be divided between A and B in the ratio

3 : 4

, but was divided

1 : 2

by mistake. Who lost, and how much?

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The same idea in a real exam question:

AFCAT · 2016 · AFCAT (II) 2016 — Numerical Ability · 04-307Moderate

Example 4 · Ratio and Proportion · Dividing in a Ratio and Partnership

An amount of ₹735 was divided between

A

,

B

and

C

. If each of them had received ₹25 less, their shares would have been in the ratio of 1 : 3 : 2. The money received by

C

was

Add the deduction back

When shares less ₹10 each are in a given ratio, the ratio describes the reduced shares. Solve for those, then add the ₹10 back before you answer.

Drill 2 more on a changed ratio, or shares with an amount taken off

Concept 5 of 5: Partnership: profit in the ratio of capital times time

Partners share profit by how much each put in and for how long. ₹10,000 for 6 months earns the same share as ₹5,000 for 12 months.

Definition

Profit shares are in the ratio of capital × months: $C_1 t_1 : C_2 t_2 : \dots$
If everyone invests for the same time, the ratio of capitals alone decides.
A partner who adds or withdraws money: add the pieces. ₹8,000 for 4 months then ₹5,000 for 8 months is $32 + 40 = 72$ (in thousands of rupee-months).
A partner who joins late invests for fewer months: joining after 3 months of a year means 9 months.

Profit ratio

P_1 : P_2 = C_1 t_1 : C_2 t_2

Worked example

A puts in ₹30,000 for a year. B joins after 4 months with ₹45,000. Out of a year's profit of ₹24,000, how much does B get?

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The same idea in a real exam question:

AFCAT · 2016 · AFCAT (I) 2016 — Numerical Ability · 04-310Easy

Example 5 · Ratio and Proportion · Dividing in a Ratio and Partnership

Dilip, Ram and Amar started a shop by investing ₹2700, ₹8100 and ₹7200 respectively. At the end of 1 yr, the profit was distributed. If Ram's share was ₹3600, their total profit was

Time counts as much as money

When partners stay for different numbers of months, the ratio of capitals alone is wrong. Multiply each capital by its months first.

Drill 5 more on partnership: profit in the ratio of capital times time

Dividing a Total in a Ratio, and Partnership

Concept 2 of 5: Dividing in a ratio of fractions

Concept 3 of 5: Numbers in a ratio with one more condition

Concept 4 of 5: A changed ratio, or shares with an amount taken off

Concept 5 of 5: Partnership: profit in the ratio of capital times time

Summary — formulas & gotchas at a glance

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