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AFCAT Numerical Ability · Mixture and Alligation

Percentage Strength of a Mixture

When a mixture is described in percentages, the liquid you do not pour in stays fixed, and its new share of the mixture gives the new total in one division.

All Mixture and Alligation notes

Why this matters

Three AFCAT past-year questions sit here, from 2016, 2018 and 2019, and four of the page's questions are HARD. Each one is a single division: the liquid that stays fixed, over its new share of the mixture. The marks go when that share belongs to the liquid being poured in.

Concept 1 of 2: When the percentage is of the liquid that stays

Whatever is not poured in stays the same amount. If the target percentage is about that liquid, it tells you the new total at once: the fixed litres are that share of it.

Definition

  • Find the litres of the liquid that is not added. It stays fixed.
  • If the new percentage is of that liquid, the new total is the fixed litres divided by the new share.
  • The amount to add is the new total minus the old one.
  • The fixed liquid can be the milk (water poured in to dilute it) or the water (milk poured in, 'make the water 30%').
  • Asked only for the new percentage after adding: fixed litres over the new total.
  • Value works the same way: milk worth a fixed sum, sold at a lower price a litre, must fill (value ÷ new price) litres; the extra is water.

New total

new total=fixed amountits new share\text{new total} = \dfrac{\text{fixed amount}}{\text{its new share}}

Worked example

How much water must be added to 30 litres of a 40% acid solution to make it 25% acid?
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

AFCAT · 2019 · AFCAT (II) 2019 — Numerical Ability · 03-201Moderate

Example 1 · Mixture and Alligation · Percentage Strength of a Mixture

A sample of 50 L of glycerine is found to be adulterated to the extent of 20%. How much pure glycerine should be added to it so as to bring down the percentage of impurity to 5%?

Divide by the new share; do not subtract percentages

Taking 16 litres of 50% milk down to 40% does not need 10% of 16 litres of water. The 8 litres of milk must be 40% of the new mixture, which is 20 litres, so add 4 litres.

A percentage is of the whole mixture

A 4% impurity means 4 litres in every 100 litres of the mixture, impurity included. It is not 4 litres for every 100 litres of the pure liquid.
Drill 21 more on when the percentage is of the liquid that stays

Concept 2 of 2: When the percentage is of the liquid you add

Pouring pure alcohol in to raise the alcohol percentage changes the alcohol, so the alcohol cannot be the fixed amount. The water is: it was one share before and is 100% minus the target after.

Definition

  • If you pour in the very liquid the percentage is about, that liquid changes. Switch to the other liquid: it stays fixed.
  • Its share after is 100% minus the target, so the new total is its litres divided by 1−p1 - p.
  • The amount to add is the new total minus the old one.
  • The equation route is a safe check: a+xT+x=p\dfrac{a + x}{T + x} = p, where aa is the liquid being added.

New total

new total=other liquid1−p\text{new total} = \dfrac{\text{other liquid}}{1 - p}

Worked example

How much pure alcohol must be added to 40 litres of a 20% alcohol solution to make it 50% alcohol?
Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

AFCAT · 2018 · AFCAT (II) 2018 — Numerical Ability · 03-204Moderate

Example 2 · Mixture and Alligation · Percentage Strength of a Mixture

75 g of a sugar solution has 30% sugar in it. The quantity of sugar that should be added to the solution to make the sugar 70% of the solution is

Using the target share for the liquid that changes

Raising 10 litres of 40% acid to 60% by adding acid: dividing the 4 litres of acid by 60% gives 6.67 litres, less than you started with. The acid is the liquid that changes. Divide the 6 litres of water by 40% instead: 15 litres, so add 5.
Drill 8 more on when the percentage is of the liquid you add

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)

Watch out for (3)

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