Problem Solving

aaron1981 wrote:The average of seven numbers is 20. The average of the first four numbers is 19 and that of the last four is 24. What is the value of the fourth number?

(A) 23
(B) 25
(C) 32
(D) 43
(E) 63

OA C

Say the seven numbers are a, b, c, d, e, f, g.

=> a + b + c + d + e + f + g = 7 x 20 = 140 ---(1)
=> a + b + c + d = 4 x 19 = 76 ---(2)
=> d + e + f + g = 4 x 24 = 96 ---(3)

Adding (2) and (3), we get,

=> (a + b + c +d + e + f + g) + d = 76 + 96 = 172 ---(4)

Plugging-in the value of (a + b + c +d + e + f + g) from (1), we get,

=> 140 + d = 172

=> [spoiler]d = 32[/spoiler]

The correct answer: C

Hope this helps!

-Jay

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Hi aaron1981,

If you're ever not sure how to handle a GMAT question, it helps to break the work down into 'steps' and work through anything that you CAN handle. Once you get some information on your pad, you'll be better able to figure out what to do next.

Here, we're told that the average of 7 numbers is 20... we can use that information to figure out the SUM of those numbers...
(Sum)/7 = 20
Sum of all 7 numbers = 140

Next, we're told that the average of the first 4 numbers is 19. We can use that information to figure out the SUM of those 4 numbers...
(Sum of 1st 4)/4 = 19
Sum of 1st 4 numbers = (4)(19) = 76

Then we're told that the average of the last 4 numbers is 24. We can use that information to figure out the SUM of those 4 numbers...
(Sum of last 4)/4 = 24
Sum of last 4 numbers = (4)(24) = 96

Now we have all of the necessary numbers we need. Notice how the "fourth number" is included in both of the sub-groups, which means if we add those two smaller sums together, the "fourth number" will be counted TWICE...

76 + 96 = 172

We know that the sum of the 7 numbers is 140, so adding that fourth number a second time increases that total to 172. Thus, that number must be 172 - 140 = 32

Final Answer: C

GMAT assassins aren't born, they're made,
Rich

The first four numbers add to 4(19) = 76. The last four numbers add to 4(24) = 96. If we add those eight numbers, we get 76+96 = 172, but we're including the middle number twice in that sum. We know if we only include the middle number once, we get 7(20) = 140, because the average of the whole set is 20. So the middle number must be 172 - 140 = 32.

aaron1981 wrote:The average of seven numbers is 20. The average of the first four numbers is 19 and that of the last four is 24. What is the value of the fourth number?

(A) 23
(B) 25
(C) 32
(D) 43
(E) 63

We can create the following equation in which variables a to g are the 1st to 7th numbers, respectively.

(a + b + c + d + e + f + g)/7 = 20

a + b + c + d + e + f + g = 140

We'll use the formula sum = average x quantity. Since the average of the first four numbers is 19, their sum is a + b + c + d = 19 x 4 = 76, and since the average of the last 4 numbers is 24, their sum is d + e + f + g = 24 x 6 = 96.

We need to determine the value of the 4th number, or d.

Adding a + b + c + d = 76 and d + e + f + g = 96, we have:

a + b + c + 2d + e + f + g = 172

Subtract a + b + c + d + e + f + g = 140 from the above equation and we have:

d = 32

Answer: C

[/spoiler]