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
In a mixture of 156 litres, the ratio of milk and water is 7 : 6. How much water should be added to make the ratio 14 : 13?
If the ratio of the first to second number is 3 : 4 and that of the second to the third number is 8 : 5, and sum of three numbers is 190 then the third number is:
The third proportional to 9 and 15 is:
The average age of 3 persons is 30 years.If their ages are in the ratio of 3 : 5 : 7 respectively, then the age of the eldest person is:
The income of A and B are in the ratio 5 : 3. The expenses of A, B and C are in the ratio of 8 : 5 : 2. If C spends 2000 and B saves ₹ 700, then A saves:
