Question

Andy and Bob took a test that had 120 problems. Andy got 30 more problems correct than Bob. The number of problems that only Andy got correct is six times the number of problems that both boys got correct. There are 20 problems that neither boy got correct. How many problems did both boys get correct?

Answer 1

Both boys get 14 problems correct of the entire test composed by 120 problems.

Procedure - Determination of the number of problems that Andy and Bob got correct in a test

In this question we must determine how many questions Andy and Bob got correct. The total of problems ($p$) is the sum of problems that neither got correct ($x$), problems that only Andy got correct ($y$) and problems that both boys got correct ($z$). Hence, we have the following algebraic expression:

$p = x + y + z$ (1)

And according to the statement, the relationships between the number of problems got correct by Andy and the number of problems that Bob got correct ($u$) is described by the following expressions:

$y = u + 30$ (2)

$y = 6\cdot z$ (3)

If we know that $p = 120$ and $x = 20$, then the solution to this system of linear equations is:

$y + z = 100$ (1)

$y - u = 30$ (2)

$y - 6z = 0$ (3)

The solution of this system is: $y = 85.714$, $z = 14.286$, $u = 55.714$.

Since the quantity of solved problems must be integers, we have the following approximate solution: $y = 86, z = 14, u = 56$

Hence, we conclude that both boys get 14 problems correct of the entire test composed by 120 problems. $\blacksquare$

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Answer 2

Answer:

the answer is 100 cuz if the test was 120 questions and both boys got 30 plus questions right and 20 questions wrong then the answer would be 100 questions right