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  1. LPU MCA Semester 1
  2. PEA515
  3. Unit 4

PEA515 · Unit 4: Ratio, proportion and mixtures

Mixture and replacement problems

30 practice questions · in the ETE syllabus

Practise this topicPractise this topic

Key points

  • Milk after n replacements = T x ((T - y)/T)^n
  • One replacement gives milk : water = (T - y) : y
  • The removed liquid contains milk in the current ratio
  • Multiply by a fraction; do not subtract y again

In a replacement problem a fixed quantity is drawn from a vessel and replaced by water (or another liquid), possibly several times. Because the mixture is uniform, each time the fraction removed is y / T of whatever milk is present.

After n replacements of y litres from T litres of pure milk:

milk left = T x ((T - y) / T)^n.

CaseResult
One replacement of y from T litres of pure milkMilk : water = (T - y) : y
Two replacementsMilk = T x ((T - y) / T)^2
Quantity replaced is unknownWrite the final milk as T times a squared fraction and solve
Mixture with ratio a : b, R litres replaced by waterMilk left = a (T - R) / (a + b); water = T - milk left

Example: from 64 litres of milk, 24 litres are drawn and replaced twice. Milk left = 64 x (40/64)^2 = 25 litres. For three replacements keeping 4/5 each time, 125 litres leaves 125 x (4/5)^3 = 64 litres.

Exam trap: the removed liquid is itself a mixture after the first step, so subtracting y from the milk each time is wrong. Multiply by the fraction instead.

Try two questions

Question 1 · Tier 3, Apply

24 litres are taken out of 64 litres of pure milk and replaced with water. The same is done again. How many litres of milk are left?

  1. 16
  2. 40
  3. 10
  4. 25
Show the answer

D. Milk left = 64 x (1 - 24/64)^2 = 64 x (40/64)^2 = 25 litres.

Why not A: Subtracts the removed quantity twice from the original.

Why not B: Counts only the first replacement.

Why not C: Subtracts the second quantity from the diluted mixture directly.

Question 2 · Tier 3, Apply

In a 56 litre drum the ratio of milk to water is 4 : 3. After 21 litres of mixture are replaced by water, what is the ratio of milk to water?

  1. 4 : 3
  2. 5 : 9
  3. 41 : 15
  4. 5 : 3
Show the answer

B. The 21 litres removed are 4/7 milk, so 20 litres of milk remain and the water is 36 litres, giving 5 : 9.

Why not A: Keeps the original ratio.

Why not C: Adds the replaced quantity to milk.

Why not D: Treats the removed quantity as the water.