AFCAT Numerical Ability · Ratio and Proportion

Mixtures and Coins

Milk and water, and bags of coins, are ratios with one twist each: when one liquid is added the other stays fixed, and a coin ratio counts coins, not rupees.

All Ratio and Proportion notes

Why this matters

None of these questions is HARD, but each has one step that is easy to get wrong: keeping the unchanged liquid fixed, or remembering that a ratio of coins is a ratio of how many coins there are. Name that step before you calculate.

Concept 1 of 5: A mixture as a ratio of two quantities

A mixture of milk and water is two quantities in a ratio. Its total splits into parts like any other total.

Definition

Milk : water $= m : w$ in a total $T$ means the milk is $\dfrac{m}{m + w}T$ and the water $\dfrac{w}{m + w}T$ .
Given the actual amounts, the ratio is their simplest form.

Milk in a mixture

\text{milk} = \dfrac{m}{m + w} \times T

Worked example

A 63-litre mixture has milk and water in the ratio

5 : 4

. How much water is in it?

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Ratio and Proportion · Mixtures and Coins

A vessel contains 90 liters of a milk-water mixture in the ratio 7:2. If 18 liters of water is added, what will be the new ratio?

The ratio is of the parts, not of the total

Milk : water

= 5 : 2

makes the milk

\tfrac 57

of the mixture. It is not

\tfrac 52

of the mixture.

Drill 3 more on a mixture as a ratio of two quantities

Concept 2 of 5: Adding one liquid: the other stays fixed

When only water is added, the milk does not change. Use the fixed quantity to work out what the other one must become.

Definition

Adding water: the milk stays at $m$ . For a new ratio $r : s$ , the water must become $\dfrac sr m$ . Subtract the water already there.
Adding milk: the water stays fixed, and the milk must become $\dfrac rs$ times the water.
With unknown amounts $3k$ and $2k$ , keep the fixed one unchanged in the new ratio: adding 10 litres of water gives $\dfrac{3k}{2k + 10} = \dfrac 34$ .
Drying works the same way: only water leaves, so the dry matter is the fixed quantity.

Water to add

\text{water needed} = \dfrac sr \times \text{milk}, \qquad \text{add} = \text{water needed} - \text{water now}

Worked example

A 45-litre mixture has milk and water in the ratio

4 : 1

. How much water must be added to make the ratio

3 : 2

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Ratio and Proportion · Mixtures and Coins

A container has milk and water in the ratio 5:3. If 16 liters of water is added, the new ratio becomes 5:4. What was the initial quantity of milk?

Adding water never changes the milk

A working that splits the new total in the new ratio has quietly changed the amount of milk. Hold the milk fixed and solve for the water.

Drill 10 more on adding one liquid: the other stays fixed

Concept 3 of 5: Replacing part of a mixture, and mixing two mixtures

Taking some mixture out removes milk and water in the same ratio as the whole. Mixing two vessels adds their milk together and their water together.

Definition

Removing part of a mixture removes the same fraction of each liquid. From 60 litres of milk and 20 of water, removing 20 litres (a quarter) leaves 45 and 15.
Then add whatever is poured back in.
Mixing: in each vessel the milk is $\dfrac{m}{m + w}$ of the volume. Add the milk from both vessels, add the water from both, then form the ratio.

Mixing two vessels

\text{milk} = \dfrac{m_1}{m_1 + w_1}V_1 + \dfrac{m_2}{m_2 + w_2}V_2

Worked example

Vessel A has milk and water in the ratio

3 : 1

, vessel B in the ratio

1 : 1

. 4 litres of A are mixed with 2 litres of B. Find the ratio of milk to water.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Ratio and Proportion · Mixtures and Coins

Two vessels are full with milk and water in the ratios 1 : 3 and 3 : 5, respectively. If both are mixed in the ratio 3 : 2, what is the ratio of milk and water in the new mixture?

Mix litres, not ratio terms

Equal amounts of a

2 : 3

and a

4 : 5

mixture do not give

(2 + 4) : (3 + 5)

. The first is split into 5 parts and the second into 9, so turn each into fractions of a litre first.

Drill 3 more on replacing part of a mixture, and mixing two mixtures

Concept 4 of 5: Coins: a ratio of how many coins there are

A coin ratio such as

2 : 3

compares how many coins there are, not what they are worth. Given the number of coins, split it first, then multiply each count by the coin's value.

Definition

₹5 and ₹10 coins in the ratio $a : b$ , $N$ coins in all: $\dfrac{a}{a + b}N$ of them are ₹5 coins.
Value is the number of coins times the face value, added over each kind of coin.

Total value

\text{value} = \sum (\text{number of coins} \times \text{face value})

Worked example

A purse has ₹1 and ₹2 coins in the ratio

5 : 3

, 64 coins in all. Find their total value.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 4 · Ratio and Proportion · Mixtures and Coins

A collection of ₹5 and ₹10 coins is in the ratio 2:3. If the total number of coins is 50, what is the total value of the coins?

Count first, then value

₹5 and ₹10 coins in the ratio

2 : 3

is a ratio of counts. Their values are in the ratio

10 : 30 = 1 : 3

Drill 5 more on coins: a ratio of how many coins there are

Concept 5 of 5: Coins when the total value is given

When the total value is given, work out what one 'set' of coins in the ratio is worth. The total is a whole number of such sets.

Definition

Coins in the ratio $a : b : c$ with face values $u, v, w$ : one set ( $a$ of the first, $b$ of the second, $c$ of the third) is worth $au + bv + cw$ .
The number of sets $k$ is the total value divided by the value of one set. Each count is its term times $k$ .
Write 50 paise as ₹0.5 and 25 paise as ₹0.25 before adding.

Number of sets

k = \dfrac{\text{total value}}{au + bv + cw}

Worked example

A bag has ₹1, ₹2 and ₹5 coins in the ratio

4 : 2 : 1

, worth ₹195 in all. How many ₹2 coins are there?

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 5 · Ratio and Proportion · Mixtures and Coins

The ratio of ₹2 and ₹5 coins in a purse is 5:8. If the total value of the coins is ₹400, what is the number of ₹5 coins?

Divide by the value of a set, not the sum of the ratio

₹1, ₹2 and ₹5 coins in the ratio

2 : 1 : 1

, worth ₹90, are

90 \div 9 = 10

sets. Dividing 90 by

2 + 1 + 1 = 4

treats every coin as worth ₹1.

Drill 6 more on coins when the total value is given

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 (5)

A mixture as a ratio of two quantities
Milk in a mixture
$\text{milk} = \dfrac{m}{m + w} \times T$
Adding one liquid: the other stays fixed
Water to add
$\text{water needed} = \dfrac sr \times \text{milk}, \qquad \text{add} = \text{water needed} - \text{water now}$
Replacing part of a mixture, and mixing two mixtures
Mixing two vessels
$\text{milk} = \dfrac{m_1}{m_1 + w_1}V_1 + \dfrac{m_2}{m_2 + w_2}V_2$
Coins: a ratio of how many coins there are
Total value
$\text{value} = \sum (\text{number of coins} \times \text{face value})$
Coins when the total value is given
Number of sets
$k = \dfrac{\text{total value}}{au + bv + cw}$

Watch out for (5)

The ratio is of the parts, not of the total
→ A mixture as a ratio of two quantities
Adding water never changes the milk
→ Adding one liquid: the other stays fixed
Mix litres, not ratio terms
→ Replacing part of a mixture, and mixing two mixtures
Count first, then value
→ Coins: a ratio of how many coins there are
Divide by the value of a set, not the sum of the ratio
→ Coins when the total value is given

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