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CDS Mathematics · Formula sheet

Ratio, Proportion and Variation formulas

11 formulas and 11 common traps for CDS Mathematics Ratio, Proportion and Variation, grouped by subtopic.

Full notes with worked examples

Combining and Dividing by Ratios

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Chaining ratios

Joining two ratios

A:B=p:q, B:C=r:s  ⇒  A:B:C=pr:qr:qsA : B = p : q,\ B : C = r : s \;\Rightarrow\; A : B : C = pr : qr : qs

Dividing a quantity in a ratio

Share in a ratio

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

Common traps

Scale, do not add

Joining 2:32 : 3 and 4:54 : 5 as 2:7:52 : 7 : 5 is wrong. Multiply each ratio so the shared term is the same number: 8:12:158 : 12 : 15.

Answer the share that was asked

When amounts are deducted before a ratio applies, the share asked for is the amount BEFORE the deduction. Solving for the reduced amount and stopping there gives a number that looks right but is not the share.

Ratios in Income, Savings and Ages

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Income, expenditure and savings

Savings

saving=income−expenditure\text{saving} = \text{income} - \text{expenditure}

Ages in a ratio

Ages then and now

M−tD−t=pq\dfrac{M - t}{D - t} = \dfrac pq

Common traps

Two ratios do not decide who saves more

Incomes 1:21 : 2 and expenses 1:31 : 3 are consistent with either person saving more, depending on the actual amounts. Try two cases before choosing.

Shift both ages

Ten years ago BOTH were ten years younger. Subtracting from only one age turns a correct setup into a wrong answer that is often printed.

Equal Ratios and Proportion Algebra

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Put each ratio equal to k

Equal ratios

ab=cd=k  ⇒  a=kb, c=kd\dfrac ab = \dfrac cd = k \;\Rightarrow\; a = kb,\ c = kd

Cross-multiply and factor

Cross-multiplication

PQ=RS  ⟺  PS=QR\dfrac PQ = \dfrac RS \iff PS = QR

Common traps

Mixed degrees cannot be determined

An expression like 3A2+4B3A−4B2\dfrac{3A^2 + 4B}{3A - 4B^2} changes with the actual size of AA and BB, not just their ratio. 'Cannot be determined' is then the right option.

Keep both signs

A quadratic in ab\dfrac ab gives two ratios, and a+ba−b\dfrac{a + b}{a - b} then takes two values of opposite sign. An option listing only the positive one is incomplete.

Direct and Inverse Variation

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Finding the constant of variation

Direct and inverse

y=kxn,xny=ky = kx^n, \qquad x^n y = k

Which quantities vary together

Test for variation

A∝B  ⟺  AB=constantA \propto B \iff \dfrac AB = \text{constant}

Common traps

Inverse means the product is constant

If yy varies inversely as x\sqrt x, then yxy\sqrt x is constant, so a bigger yy needs a SMALLER xx. An answer larger than the starting xx signals a direct-variation slip.

Count the degree

After x=kyx = ky, an expression of degree nn becomes a constant times yny^n. So x3y4\dfrac{x^3}{y^4} (degree −1-1) varies inversely as yy, not directly.

Partnership

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Capital × time

Profit share

share∝capital×time\text{share} \propto \text{capital}\times\text{time}

Common traps

Count each period separately

A partner who withdraws part of the capital after four months holds the full amount for four months and the rest for eight. Using the final capital for the whole year understates the share.

Mixtures and Alligation

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Combining mixtures

Component in a mixture

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

Alligation

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

Common traps

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.

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.

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