In the correctly worked addition problem shown, P, Q, R, and S are digits.

BTGmoderatorDC wrote: ↑

Fri Nov 18, 2022 4:35 am

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In the correctly worked addition problem shown, P, Q, R, and S are digits. If Q = 2P, which of the following could be the value of S ?

(A) 3
(B) 2
(C) 1
(D) -1
(E) -2


OA A

Source: Official Guide

For this question, it's best to use number sense.
However, we can also solve this question algebraically

Key property: The VALUE of the 3-digit integer abc can be written as follows: abc = 100a + 10b + c
For example: 625 = 100(6) + 10(2) + 5

Likewise: 3P5 = 100(3) + 10(P) + 5, 4QR = 100(4) + 10(Q) + R, and 8S4 = 100(8) + 10(S) + 4

So the addition, 3P5 + 4QR = 8S4 can be written algebraically as: (300 + 10P + 5) + (400 + 10Q + R) = 800 + 10S + 4
Simplify: 705 + 10P + 10Q + R = 804 + 10S
Subtract 705 from both sides to get: 10P + 10Q + R = 99 + 10S
Subtract 10S from both sides: 10P + 10Q - 10S + R = 99
Subtract R from both sides: 10P + 10Q - 10S = 99 - R
Factor the left side: 10(P + Q - S) = 99 - R
Finally, since we're told that Q = 2P, we can substitute to get: 10(P + 2P - S) = 99 - R
Simplify: 10(3P - S) = 99 - R
Since 10(3P - S) is a multiple of 10, and since R is a digit, we know that R must equal 9
We get: 10(3P - S) = 99 - 9 = 90

Now take: 10(3P - S) = 90
Divide both sides by 10 to get: 3P - S = 9
Rearrange to get: S = 3P - 9
Factor to get: S = 3(P - 3)
This means S is a multiple of 3

Check the answer choices..... answer choice A (3) is the only multiple of 3

Answer: A

Cheers,
Brent