Two numbers are in the ratio 3 : 5. If 9 is subtracted from the numbers, the ratio becomes 12 : 23. The numbers are
33, 55
The standard method for a ratio problem is to express both quantities in terms of a single unknown, so that the second condition becomes an equation in that unknown.
Since the numbers are in the ratio 3 to 5, write them as 3x and 5x for some positive number x. This automatically satisfies the first condition, whatever x turns out to be.
Subtracting 9 from each gives 3x minus 9 and 5x minus 9, and the second condition says these are in the ratio 12 to 23. So (3x - 9) divided by (5x - 9) equals 12 divided by 23.
Cross multiplying gives 23(3x - 9) = 12(5x - 9), that is 69x - 207 = 60x - 108. Collecting terms, 69x - 60x = 207 - 108, so 9x = 99 and x = 11.
The numbers are therefore 3 times 11 and 5 times 11, which is 33 and 55.
Checking confirms it: 33 and 55 are in the ratio 3 to 5, and subtracting 9 from each leaves 24 and 46, which reduce to 12 to 23 on dividing by 2.
The alternative route, useful when time is short, is to test the options directly, since every option already satisfies the 3 to 5 ratio. Subtracting 9 from 30 and 50 gives 21 and 41, which is not 12 to 23. From 36 and 60 it gives 27 and 51, which reduces to 9 to 17. From 42 and 70 it gives 33 and 61. Only 33 and 55 works.
Hence the numbers are 33 and 55.
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Important Questions from Ratio and Proportion
- A box containing the denominations of one rupee coins, five rupee coins and ten rupee coins amount to Rs.4320. There are equal number of
coins in all denominations. What is the total number of coins in the box? - The sum of three terms is 14380. The ratio between first term and second term is 14:15 and the ratio between second term and the third term is 16:17. Find the difference between first and last term.
- A box containing the denominations of fifty paise coins, two rupee coins and five rupee coins amount to Rs. 1275. There are equal number of coins in all denominations. What is the value of two rupee coins in the box? (In Rs.)
- Three numbers are in the ratio 19:21:23. If the difference between thrice the third number and the sum of first and second number is 841. Find the difference between the first and third number?
- The ratio between the number of males and females in a college is 27:28. If the number of males are increased by 20% and the number of females are increased by 30%, then what will the new ratio of males and females in the college?
