emmak wrote:
A bank offers an interest of 5% per annum compounded annually on all its deposits. If $10,000 is deposited, what will be the ratio of the interest earned in the 4th year to the interest earned in the 5th year?
A. 1:5
B. 625:3125
C. 100:105
D. 100^4:105^4
E. 725:3225
VERITAS PREP OFFICIAL SOLUTION:Solution: (C)
This is a great example of a problem that looks much more difficult than it really is. If we calculate the balance of this investment year-to-year, it would be:
First year: 10,000 + 5
NOTE: Using fractions is typically the easiest way to calculate, so we’ll represent 5% as 1/20 from here on out.
Second year: \(10,000∗\frac{21}{20}+\frac{1}{20}∗(10,000∗\frac{21}{20})=\frac{21}{20}(10,000∗\frac{21}{20})=(\frac{21}{20})^2∗10,000\)
Third year: \((\frac{21}{20})^2(10,000)+\frac{1}{20}∗(\frac{21}{20})^2(10,000)=\frac{21}{20}∗(\frac{21}{20})^2(10,000)=(\frac{21}{20})^3∗10,000)\)
If you follow the pattern, the total value at the end of each year will simply be \((\frac{21}{20})^n(10,000)\) at the end of the nth year. The amount of interest each year is 1/20 of the previous year’s balance (that …+1/20 * the previous year). So, the amount of interest calculated in the 4th year will be: \(\frac{1}{20}∗(\frac{21}{20})^3(10,000)\)
And the amount of interest earned in the 5th year will be: \(\frac{1}{20}∗(\frac{21}{20})^4(10,000)\)
Putting those into ratio, you’ll see that the 1/20 and the 10,000 is common to both, so those terms divide out, leaving simply: \(\frac{(\frac{21}{20})^3}{(\frac{21}{20})^4}\)
Factoring out the common \((\frac{21}{20})^3\) term, we’re left with 1/(21/20). Dividing by a fraction is the same as multiplying by the reciprocal, so that can be expressed as 20/21, which is the same as 100/105. Therefore, the correct answer is C.
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