In September, the incomes of Kamal, Amal and Vimal are in the ratio 8:6:5. They rent a house together, and Kamal pays 15%,Amalpays12\%andVimalpays18\%oftheirrespectiveincomestocoverthetotalhouserentinthatmonth.InOctober,thehouserentremainsunchangedwhiletheirincomesincreaseby10\% , 12\%and15\% , respectively. In October, the percentage of their total income that will be paid as house rent, is nearest to
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Official Correct Answer: A. First, the question says that in September, their incomes are in the ratio 8:6:5. That means if I let Kamal's income be 8 parts, Amal's is 6 parts, and Vimal's is 5 parts. We represent their incomes as 8x, 6x, and 5x respectively, where x is some common multiple.
Next, it mentions that they pay a certain percentage of their incomes as rent. Kamal pays 15%, Amal pays 12%, and Vimal pays 18%. So, We calculate how much each person pays in rent.
Let me write that down:
- Kamal's rent: 15% of 8x = 0.15 * 8x = 1.2x
- Amal's rent: 12% of 6x = 0.12 * 6x = 0.72x
- Vimal's rent: 18% of 5x = 0.18 * 5x = 0.9x
Now, the total rent paid in September would be the sum of these amounts. So, adding them up:
Total rent = 1.2x + 0.72x + 0.9x = (1.2 + 0.72 + 0.9)x = 2.82x
Okay, so the total rent is 2.82x in September.
Moving on to October, their incomes increase by 10%, 12%, and 15% respectively. We calculate their new incomes first.
- Kamal's new income: 8x increased by 10% = 8x * 1.10 = 8.8x
- Amal's new income: 6x increased by 12% = 6x * 1.12 = 6.72x
- Vimal's new income: 5x increased by 15% = 5x * 1.15 = 5.75x
So, their new incomes are 8.8x, 6.72x, and 5.75x.
The house rent remains the same, which is 2.82x. Now, We find out what percentage of their total income in October is spent on rent.
First, Calculating their total income in October:
Total income in October = 8.8x + 6.72x + 5.75x
Summing these: 8.8x + 6.72x = 15.52x
15.52x + 5.75x = 21.27x
So, total income is 21.27x.
Now, the rent is 2.82x, so the percentage of income spent on rent is:
Percentage = (Rent / Total Income) * 100 = (2.82x / 21.27x) * 100
Wait, the x cancels out, so it's (2.82 / 21.27) * 100.
Compute that.
First, 2.82 divided by 21.27. Hmm, let me do this division step by step.
21.27 goes into 2.82 how many times? Since 21.27 is larger than 2.82, it's less than 1. Compute 2.82 / 21.27.
I can write this as 282 / 2127 to make it easier. if I can simplify this fraction.
Divide numerator and denominator by 3:
282 ÷ 3 = 94
2127 ÷ 3 = 709
So, 94/709. Hmm, that doesn't seem to simplify further. Compute this division.
94 ÷ 709. Let me approximate this.
709 goes into 940 once (since 709*1=709). Subtract 709 from 940: 940 - 709 = 231.
Bring down a zero: 2310. 709 goes into 2310 three times (709*3=2127). Subtract: 2310 - 2127 = 183.
Bring down another zero: 1830. 709 goes into 1830 two times (709*2=1418). Subtract: 1830 - 1418 = 412.
Bring down another zero: 4120. 709 goes into 4120 five times (709*5=3545). Subtract: 4120 - 3545 = 575.
Bring down another zero: 5750. 709 goes into 5750 eight times (709*8=5672). Subtract: 5750 - 5672 = 78.
So, putting it all together, 94 ÷ 709 ≈ 0.1326...
0.1326 * 709 ≈ 94?
0.1 * 709 = 70.9
0.03 * 709 = 21.27
0.002 * 709 = 1.418
0.0006 * 709 ≈ 0.4254
Adding these up: 70.9 + 21.27 = 92.17; 92.17 + 1.418 = 93.588; 93.588 + 0.4254 ≈ 94.0134. So, 94/709 ≈ 0.1326, which is 13.26%.
Therefore, the percentage of their total income spent on rent in October is approximately 13.26%, which is option A.