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[Smart Math] Averages Problem 11

Here’s and example of a SMART MATH problem for AVERAGES.

Problem

The average temperature for the first 9 days was 20 ${}^\circ$C and that of the first 5 was 25 ${}^\circ$C and that for the last 5 days was 15 ${}^\circ$C. What was the temperature on the fifth day?

1. 25 ${}^\circ$C
2. 20 ${}^\circ$C
3. 30 ${}^\circ$C
4. 15 ${}^\circ$C
5. 27 ${}^\circ$C

The Usual Method

Total temperature for 9 days = 9 x 20 = 180 ${}^\circ$C

Total temperature for first 5 days = 5 x 25 = 125 ${}^\circ$C

Total temperature for last 5 days = 5 x 15 = 75 ${}^\circ$C

Also, total temperature for first 5 days = a + b

Where, a = total temperature for first 4 days and

b = temperature on the 5th day

Similarly, total temperature for last 5 days = b + c

Where, b = temperature on the 5th day and

c = total temperature for last 4 days

$\therefore a+b=125$                            …… Eq. 1.

and $b+c=75$                            …… Eq. 2.

Also $a+b+c=180$

Adding Eq. 1 and Eq. 2, we get: a + 2b + c = 200 ${}^\circ$C

$\therefore a+b+c+b=200$

$\therefore 180+b=200$

$\therefore b=$20 ${}^\circ$C

Hence, temperature on 5th day = 20 ${}^\circ$C

(Ans: 2)

Estimated Time to arrive at the answer = 45 seconds.

Using Technique

Since the average temperatures for the first 5 days and the last 5 days is ‘equidistant’ from the average temperature for the 9 days, we can simply take the arithmetic mean of 25 and 15 as $\frac{25+15}{2}=\frac{40}{2}=$ 20 ${}^\circ$C as the temperature of the 5th day.