The ratio of monthly incomes of A and B is 7 : 10. The ratio of their expenditures is 2 : 3. If each of A and B saves ₹1,000 per month, then what will be the monthly income of B?

Correct — B, ₹10,000. Write the incomes as 7x and 10x and the expenditures as 2y and 3y, since a ratio fixes only the proportion and not the amount. Savings are income minus expenditure, so 7x − 2y = 1000 and 10x − 3y = 1000. Multiply the first by 3 and the second by 2 to match the y terms: 21x − 6y = 3000 and 20x − 6y = 2000. Subtracting gives x = 1000, and substituting back gives 2y = 7000 − 1000 = 6000, so y = 3000. B's monthly income is 10x = ₹10,000. The figures check out: A earns 7,000 and spends 6,000, B earns 10,000 and spends 9,000, and each is left with 1,000.

• (a)₹9,000 — This is B's expenditure, 3y = 9,000. It is the value you land on if you solve for y and then read the wrong quantity off the page.
• (c)₹15,000 — Would make A's income 10,500 and force a savings figure well above 1,000 for any expenditure ratio of 2 : 3. The two equations have a unique solution and this is not it.
• (d)₹12,000 — Fits neither equation. With B at 12,000 the common multiplier x would be 1,200, A's income 8,400, and the expenditure ratio would no longer come out as 2 : 3 with equal savings.

A ratio is not a set of values. Two ratios given in one question need two different multipliers — x for the incomes, y for the expenditures — and the sentence that links them, here the equal saving, supplies the equations. Two unknowns need two equations, and this stem provides exactly two.

There is a faster route worth knowing. Savings are equal, so 7x − 2y = 10x − 3y, which collapses at once to y = 3x. Substituting into 7x − 2y = 1000 gives 7x − 6x = 1000, so x = 1000 and B's income is 10,000 in two lines. That trick — set the two expressions for the same quantity equal to each other before using its value — works whenever a problem says two people save the same amount, or spend the same amount, or are left with the same balance.

• Income = expenditure + savings; a ratio needs its own multiplier before it can be used.
• Incomes 7x and 10x, expenditures 2y and 3y, with 7x − 2y = 10x − 3y = 1000.
• Equating the two savings gives y = 3x directly.
• x = 1000 and y = 3000, so the incomes are ₹7,000 and ₹10,000 and the expenditures ₹6,000 and ₹9,000.
• Check every answer by rebuilding the story: both parties must save exactly ₹1,000.

• Using the same multiplier for the income ratio and the expenditure ratio.
• Solving correctly and then reporting A's income, or B's expenditure, instead of B's income.
• Assuming the ratio of savings must equal the ratio of incomes; here the savings are equal while the incomes are not.

A two-ratio item solved by one linking sentence. Almost every version of it says the two people save, or spend, the same amount.

• practice — not a real PYQ

  The incomes of A and B are in the ratio 4 : 5 and their expenditures in the ratio 3 : 4. If each saves ₹2,000, A's income is

  • (a)₹8,000
  • (b)₹10,000
  • (c)₹12,000
  • (d)₹6,000

  Answer(a) ₹8,000 — equal savings give 4x − 3y = 5x − 4y, so y = x; then 4x − 3x = 2000 and A earns 4x = 8,000.

• practice — not a real PYQ

  A person's income rises by 20% and his expenditure by 10%. If he earlier saved a fifth of his income, his savings now rise by

  • (a)10%
  • (b)20%
  • (c)50%
  • (d)60%

  Answer(d) 60% — take income 100 and expenditure 80; new savings are 120 − 88 = 32 against 20 earlier.