A2D2
Why are multiplying with 1.08 and not 0.08
Isn’t 8% is 0.08
Hi
A2D2,
Thanks for your query.
Please observe that though the interest rate is 8%, which definitely equals 0.08, the question does not just ask for the INTEREST earned on ‘x’. Observe that the question is about the relationship between x and w, where w is the AMOUNT of the investment after 2 years. And Amount is composed of two parts:
Principal + Interest.So, we need to find the amount at the end of year 1 (to find the principal for year 2) and then, the amount at the end of year 2.
Read on to find exactly how 1.08 is the right factor to multiply here, and not just 0.08.
DETAILED EXPLANATION: Alex initially deposited ‘x’ dollars at an 8% interest rate, compounded annually.
So,
Interest earned during the first year = 8% of x =
0.08x Hence, the
amount in Alex’s account after one year - = Initial investment + Interest earned
- = x + 0.08x = x * (1 + 0.08) = 1.08x …(I) [Observe how we got 1.08 here when we added the initial investment to the interest earned. You had stopped at the interest alone. 😊]
Now, at the beginning of the second year, Alex again deposited additional x dollars in the same account. So, the
total principal at the beginning of 2nd year =
final amount of the end of first year + New investment.
- Using (I), we can say, Principal at the beginning of 2nd year = 1.08x + x …(II)
Now, the final question: Amount in Alex’s account at the end of 2nd year.
Since the account still adds an annual interest of 8%, and the principal at the beginning of the second year now becomes (1.08x + x), Alex will get an interest of 8% on this new principal.
- So, INTEREST earned during second year = (1.08x + x) × 0.08
- And the FINAL AMOUNT (that is, w) after the second year = Principal + Interest
- That is, w = (1.08x + x) + [(1.08x + x) × 0.08]
- w = (1.08x + x) (1 + 0.08) [Taking (1.08x + x) common from both terms]
- = 1.08 * (1.08x + x)
Observe carefully how for both years, the interest of
0.08 becomes
1.08 because of adding the principal amount. This is exactly what confused you!
Hope this will clarify!
Best,
Aditi Gupta
Quant expert,
e-GMAT