Aptitude · Ratio and Proportion · Model 9

Two ratios, two scales, and one line joining them

Q: Give the cross-product shortcut for that.

(i₁e₂ − i₂e₁) income parts are worth (e₂s₁ − e₁s₂) . Here (5×4) − (6×3) = 2 parts and (4×1800) − (3×1600) = 2400, so one part is ₹1,200. It is the elimination done in advance.

An income ratio and an expenditure ratio have no reason to share a scale, so they need two separate unknowns. The one sentence that connects them — income minus expenditure is savings — turns that into two equations, and a cross-product shortcut solves them in a line.

Solve a two-person budget from both ratios →

Income − expenditure = savings, applied to each person. The income ratio gets its own unknown and the expenditure ratio gets another; using one letter for both is the standard error.

01 The idea

Why one unknown is not enough

Two people have incomes in the ratio 5 : 6 and expenditures in the ratio 3 : 4. They save ₹1,800 and ₹1,600. Find the second person’s income. The temptation is to write the incomes as 5x and 6x and the expenditures as 3x and 4x, and it is wrong — there is no reason an income part and an expenditure part should be worth the same amount. Two ratios, two unknowns: incomes 5k and 6k, expenditures 3m and 4m.

One sentence joins them: income minus expenditure is savings. Applied to each person that gives 5k − 3m = 1800 and 6k − 4m = 1600 — two equations in two unknowns, solvable by elimination. Multiply the first by 4 and the second by 3 and the m terms match: 20k − 12m = 7200 and 18k − 12m = 4800, so 2k = 2400 and k = 1200.

The second person’s income is 6k = ₹7,200. Back-substituting gives 3m = 5(1200) − 1800 = 4200, so m = 1400 — a different number from k, which is the point. The full picture is incomes of ₹6,000 and ₹7,200, expenditures of ₹4,200 and ₹5,600, and savings of ₹1,800 and ₹1,600 as stated.

That elimination has been done once and for all, which gives the shortcut this model is known for. Cross-multiply the income ratio against the expenditure ratio to get a number of income parts, and cross-multiply the expenditure ratio against the savings to get what those parts are worth: (5×4) − (6×3) = 2 parts, and (4×1800) − (3×1600) = 2400. Two parts are ₹2,400, so one is ₹1,200. Same answer, one line.

Income minus expenditure is savings. Give the income ratio one unknown and the expenditure ratio another, then write that sentence twice.

The identityIncome − Expenditure = Savings. Rearranged as needed: expenditure is income minus savings, and income is expenditure plus savings. Every question in this model is one of those three.

Two scalesThe income parts are worth k each and the expenditure parts m each, and k ≠ m in general. Here k = ₹1,200 and m = ₹1,400.

The double cross(i₁e₂ − i₂e₁) income parts are worth (e₂s₁ − e₁s₂). On 5 : 6, 3 : 4 with savings 1800 and 1600 that is 2 parts worth ₹2,400.

02 Worked example

Incomes 5 : 6, expenditures 3 : 4, savings ₹1,800 and ₹1,600

One pair of household budgets runs the whole lesson. A and B have monthly incomes in the ratio 5 : 6 and monthly expenditures in the ratio 3 : 4. If they save ₹1,800 and ₹1,600 respectively, find B’s monthly income.

Give the two ratios two different unknownsAn income part and an expenditure part are different amounts of money. Writing 3x and 4x for the expenditures would assert that they are equal, which the data never said.incomes 5k and 6k expenditures 3m and 4m

Write the identity twiceOne equation per person. Each one mixes k and m, which is why neither can be solved alone.5k − 3m = 1800 6k − 4m = 1600

Eliminate mMultiply the first equation by 4 and the second by 3 so both carry 12m, then subtract.20k − 12m = 7200, 18k − 12m = 4800 ⇒ 2k = 2400

Read off the income, then recover mOne income part is ₹1,200, so B’s income is 6 of them. Back-substitute for the expenditure part, and note that it is a different number.k = 1200 ⇒ B = ₹7,200 3m = 6000 − 1800 = 4200 ⇒ m = 1400

Fill in the table and check all three factsBoth savings must reappear and both ratios must survive. Three checks, and together they catch every error this model can produce.6000 − 4200 = 1800 ✓ 7200 − 5600 = 1600 ✓ 6000 : 7200 = 5 : 6 ✓ 4200 : 5600 = 3 : 4 ✓

Look at k = 1,200 against m = 1,400. Those are the two numbers that a single-letter setup asserts are equal, and they are not close. If you ever get an answer here where the two scales come out the same, that is worth a second look — it happens only when the savings themselves are in the same ratio as something else in the problem, and it is much more often a sign that you used one letter twice.

03 The method

The six shapes this model comes in

The source treats this as a masterclass in phases, and the phases are worth knowing by name because each has its own shortcut.

Income − Expenditure = Savings. With incomes i₁k, i₂k and expenditures e₁m, e₂m: (i₁e₂ − i₂e₁) k = e₂s₁ − e₁s₂, then back-substitute for m. For one person alone, income is just expenditure plus savings, so an expenditure-to-savings ratio of 26 : 3 makes the income 29 parts.

When the savings are equal, skip the algebra entirely. Line the two ratios up vertically and scale the income ratio until both vertical drops are the same number of parts — that shared drop is the saving. Incomes 8 : 7 against expenditures 13 : 11: double the income ratio to 16 : 14, and now 16 drops to 13 and 14 drops to 11, both by 3 parts. So 3 parts = ₹12,000 and one part is ₹4,000, making the first income 16 × 4000 = ₹64,000. And watch for the annual trap: a question about monthly salaries that asks for an annual figure wants your answer multiplied by 12, and the un-multiplied number is always one of the options.

Shape	What you are given	Method	Example answer
One person	expenditure : savings = 26 : 3, income ₹7,250	`income = 29 parts`	savings ₹750
The annual trap	salaries 2 : 3 : 5, C is ₹12,000 above A	`×12 at the end`	B’s annual ₹1,44,000
Equal savings, matching drops	incomes 3 : 4, expenditures 2 : 3, save 6,000 each	`drop is 1 part each`	P’s income ₹18,000
Equal savings, scaling needed	incomes 8 : 7, expenditures 13 : 11, save 12,000 each	`double the income ratio`	₹64,000
Unequal savings	incomes 5 : 6, expenditures 3 : 4, save 1,800 and 1,600	`the double cross`	B’s income ₹7,200
Three-person cascade	incomes A : B = 5 : 3, expenditures 8 : 5 : 2, C spends 2,000, B saves 700	`walk C → B → A`	A saves ₹1,500
Percentage merged in	incomes 15 : 7 : 9, expenditures 12 : 6 : 5, X saves 1/3 of income	`force X to 3 : 2`	savings 30 : 12 : 29
Data that contradicts itself	incomes 5 : 4, expenditures 3 : 2, save 800 and 600	`gives a negative expenditure`	no such pair exists

05 Cheat sheet

Model 9 on one page

One identity, three methods and the two traps — the annual multiplier and the self-contradicting data set.

Case	Rule	On the lesson numbers
The identity	`I − E = S`	6000 − 4200 = 1800
Two ratios	`i₁k, i₂k and e₁m, e₂m`	k = 1,200 and m = 1,400
The double cross	`(i₁e₂ − i₂e₁)k = e₂s₁ − e₁s₂`	2k = 2,400
Equal savings	`match the vertical drops`	8 : 7 doubled to 16 : 14 against 13 : 11
One person only	`income = E parts + S parts`	26 : 3 gives 29 parts
Monthly asked as annual	`multiply by 12`	₹12,000 becomes ₹1,44,000
One letter for both ratios	`the standard error`	asserts 1,200 = 1,400

The two scales are genuinely differentHere an income part is ₹1,200 and an expenditure part is ₹1,400. Any setup that uses x for both is asserting they are equal, and it will produce a wrong answer that still satisfies one of the two ratios.

Equal savings is a shortcut, not a special caseWhen both people save the same amount, scale the income ratio until both vertical drops to the expenditure ratio are the same number of parts. That drop is the saving, and the whole question becomes one division.

Some published questions are impossibleIncomes 5 : 4 with expenditures 3 : 2 and savings of ₹800 and ₹600 gives an expenditure part of −₹100. The cross-product shortcut still returns a plausible number, which is how such questions get published with answers — so check that your expenditures come out positive.

06 Where & why

Where this shows up

This is the highest-value model in the chapter for bank and government papers, and it is where the chapter starts paying for the two-unknown habit.

Bank PO · Clerk

“If both save ₹12,000, find A’s income”

The equal-savings shape, set constantly. The vertical-drop method answers it in about twenty seconds once you can see which ratio needs scaling.

SSC CGL · Mains

Unequal savings and the savings ratio

The double cross for the two-person version; for the three-person version with a stated savings percentage, force one person’s income-to-expenditure ratio to match the percentage and scale both lines to suit.

The annual trap

“… what is B’s annual salary?”

Monthly salaries in the stem, an annual figure in the question, and the monthly answer sitting in the options. Worth reading the last four words of every question in this model twice.

Real budgeting

Savings rate against income

The identity is the actual arithmetic of a household budget, and the useful reading is that a 5 : 4 income-to-expenditure ratio is a 20% savings rate. That conversion between a ratio and a percentage is the bridge to the percentages module.

Two habits carry this model. Use two letters, always, even when the ratios look compatible. And check that the expenditures come out positive before you write the answer down, because the shortcut will not tell you when they do not.

07 Interview questions

What gets asked

Twelve, because this is the model with the most distinct shapes and the one most worth rehearsing out loud.

State the identity this whole model rests on.

Income minus expenditure equals savings. Rearranged, expenditure is income minus savings and income is expenditure plus savings. Every question in the model is that sentence applied to one person or to each of two or three.

Why do you need two unknowns rather than one?

Because the income ratio and the expenditure ratio have no reason to share a scale. Writing incomes as 5x and 6x and expenditures as 3x and 4x asserts that an income part and an expenditure part are the same amount of money. In the lesson example they are ₹1,200 and ₹1,400.

Incomes 5 : 6, expenditures 3 : 4, savings ₹1,800 and ₹1,600. Find B’s income.

₹7,200. With incomes 5k, 6k and expenditures 3m, 4m the two equations are 5k − 3m = 1800 and 6k − 4m = 1600. Eliminating m gives 2k = 2400, so k = 1200 and B’s income is 6k.

Give the cross-product shortcut for that.

(i₁e₂ − i₂e₁) income parts are worth (e₂s₁ − e₁s₂). Here (5×4) − (6×3) = 2 parts and (4×1800) − (3×1600) = 2400, so one part is ₹1,200. It is the elimination done in advance.

A man’s expenditure and savings are in the ratio 26 : 3 and his income is ₹7,250. Find his savings.

₹750. For one person the income is expenditure plus savings, so it is 29 parts. 7250 ÷ 29 = ₹250 a part, and savings are 3 parts. A quick check on the options: the answer must be a multiple of 3.

Monthly salaries are in the ratio 2 : 3 : 5 and C earns ₹12,000 more than A. Find B’s annual salary.

₹1,44,000. The gap is 3 parts, so one part is ₹4,000 and B’s monthly salary is 3 parts = ₹12,000. The question asks for the annual figure, so multiply by 12. Note that ₹12,000 will be one of the options, and it is the wrong one.

Incomes 3 : 4, expenditures 2 : 3, and both save ₹6,000. Find P’s income.

₹18,000. Line the ratios up: 3 drops to 2 and 4 drops to 3, both by exactly one part. So one part is the saving, ₹6,000, and P’s income is 3 parts. No algebra needed because the drops already matched.

Incomes 8 : 7, expenditures 13 : 11, and both save ₹12,000. Find the first income.

₹64,000. The drops do not match, so scale the income ratio: double it to 16 : 14. Now 16 drops to 13 and 14 drops to 11, both by 3 parts. Three parts are ₹12,000, so one is ₹4,000 and the first income is 16 × 4,000.

Income A : B = 5 : 3 and expenditure A : B : C = 8 : 5 : 2. C spends ₹2,000 and B saves ₹700. Find A’s savings.

₹1,500. Walk the chain: C’s 2 expenditure parts are ₹2,000, so an expenditure part is ₹1,000 and A and B spend ₹8,000 and ₹5,000. B’s income is 5,000 + 700 = ₹5,700, which is 3 income parts, so an income part is ₹1,900 and A earns ₹9,500. A’s savings are 9,500 − 8,000.

Why does that cascade have to be walked in that order?

Because only one absolute number is given for each ratio, and they are given for different people. C is the only person with a stated amount in the expenditure ratio, so the expenditure scale has to come from C; B is the only person who then has both an expenditure and a savings figure, so the income scale has to come from B. There is no other order that works.

Can a question in this model be impossible?

Hard

(a) The incomes of A and B are in the ratio 5 : 3 and the expenditures of A, B and C are in the ratio 8 : 5 : 2. C spends ₹2,000 and B saves ₹700. Find A’s savings. (b) Explain why C’s savings cannot be found from this data. (c) A different question states that two people have incomes in the ratio 5 : 4, expenditures in the ratio 3 : 2, and savings of ₹800 and ₹600 respectively. Show that no such pair of people can exist, and state which quantity gives the contradiction away.

Follow-up

Part (a) is a chain that can only be walked in one order, part (b) tests whether you notice that C never appears in the income ratio at all, and part (c) is a published practice question with a published answer that is impossible. Being able to prove a question wrong is a real skill, and the quantity that betrays it is the expenditure, which comes out negative.

Show the hint

For (a) start from the only stated amount, C’s expenditure. For (c) solve properly with two unknowns rather than using the cross-product shortcut, and look at what the expenditure part comes out to.