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CDS Mathematics · Ratio, Proportion and Variation

Mixtures and Alligation

Add up each component separately when mixtures are combined; use alligation to find the ratio in which two mixtures give a target mixture.

All Ratio, Proportion and Variation notesCDS Mathematics strategy

Why this matters

Thirteen PYQs. When the amounts are given, track each component (the copper, the milk) on its own and divide at the end. When a target strength is given, alligation gives the mixing ratio in one line — work with the fraction of ONE component throughout.

Concept 1 of 2: Combining mixtures

Mixing never creates or destroys milk or water. So compute how much of each component every part contributes, add them, and form the new ratio. Adding pure water changes the total but leaves the other component fixed.

Definition

  • From WW kg of an alloy in the ratio a:ba : b, the first metal is aa+bW\dfrac{a}{a + b}W.
  • Equal quantities: add the fractions directly.
  • Adding water (or one pure component) keeps the other component's AMOUNT fixed; set its new fraction and solve for the total.
  • A component's share can only move toward the added substance: adding kerosene cannot lower the kerosene share.

Component in a mixture

amount of first=aa+b×quantity\text{amount of first} = \dfrac{a}{a + b}\times\text{quantity}

Worked example

30 kg of an alloy with copper and tin 1:21 : 2 is melted with 40 kg of an alloy with copper and tin 3:53 : 5. Find the new ratio.
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2017 · CDS (I) 2017 — Elementary Mathematics · Q50Moderate

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

25 kg of alloy X is mixed with 125 kg of alloy Y. If the amount of lead and tin in the alloy X is in the ratio 1:21 : 2 and the amount of lead and tin in the alloy Y is in the ratio 2:32 : 3, then what is the ratio of lead to tin in the mixture ?

Do not average the ratios

Mixing 1:21 : 2 with 2:32 : 3 does not give 3:53 : 5 unless the amounts are chosen specially. Work out the actual quantity of each component first.
Drill 7 more on combining mixtures

Concept 2 of 2: Alligation

Mixing a weak and a strong solution gives something in between. The closer the target is to one of them, the more of that one you need — the mixing ratio is the two DISTANCES from the target, crossed over.

Definition

  • With strengths c1<c<c2c_1 < c < c_2 (fractions of one component), the quantities are in the ratio first:second=(c2−c):(c−c1)\text{first} : \text{second} = (c_2 - c) : (c - c_1).
  • Use the fraction of the SAME component everywhere (spirit fraction 15\dfrac15, not the ratio 1:41 : 4).
  • Three components: if one component comes from only one source (tin only in Y), use it to find the mixing ratio first.

Alligation

firstsecond=c2−cc−c1\dfrac{\text{first}}{\text{second}} = \dfrac{c_2 - c}{c - c_1}

Worked example

One solution is 20%20\% sugar and another 50%50\%. In what ratio should they be mixed to get 30%30\%?
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2026 · CDS (II) 2026 — Elementary Mathematics · Q25Moderate

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

An alloy (X) of gold and silver is taken in the ratio 1 : 2 and another alloy (Y) of gold and silver is taken in the ratio 2 : 3. How many parts of X and Y must be taken to obtain a new alloy Z consisting of gold and silver in the ratio 4 : 7 ?

Cross the distances

The WEAK solution's share is the distance from the target to the STRONG one. Putting each distance next to its own solution gives the reciprocal ratio.
Drill 4 more on alligation

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)

  • Combining mixtures

    Component in a mixture

    amount of first=aa+b×quantity\text{amount of first} = \dfrac{a}{a + b}\times\text{quantity}
  • Alligation

    Alligation

    firstsecond=c2−cc−c1\dfrac{\text{first}}{\text{second}} = \dfrac{c_2 - c}{c - c_1}

Watch out for (2)

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 testDrill all 13 questions