Question

The sum of two numbers is $ - 16 $ . Three times the larger equals the smaller. How do you find the larger number?

Answer 1

Hint: In order to determine the larger number, we will consider the larger number as $ a $ and the smaller number as $ b $ . Then, form two equations using the clues given in the question. Then we will derive $ a $ from the second equation and substitute it in the first equation, so as to determine the smaller number i.e., $ b $ . And, we will substitute the value of $ b $ in the second equation to determine the value of $ a $ i.e., the larger number.

Complete step-by-step answer:
Now, let us consider the larger number as $ a $ and the smaller number as $ b $ .
It is given that the sum of two numbers is $ - 16 $ .
Thus, we have,
 $ a + b = - 16 $ $ \to \left( 1 \right) $
It is also given that three times the larger equals the smaller.
 $ 3a = b $ $ \to \left( 2 \right) $
 $ a = \dfrac{b}{3} $
Let us substitute this value in equation $ \left( 1 \right) $ , we have,
 $ \dfrac{b}{3} + b = - 16 $
 $ \dfrac{{b + 3b}}{3} = - 16 $
 $ b + 3b = - 16 \times 3 $
 $ 4b = - 48 $
 $ b = \dfrac{{ - 48}}{4} $
Hence, $ b = - 12 $
Therefore, the smaller number is $ - 12 $ .
Now, substitute the value of $ b $ in the equation $ \left( 2 \right) $ , we have,
 $ 3a = - 12 $
 $ a = \dfrac{{ - 12}}{3} $
Hence, $ a = - 4 $
Therefore, the larger number is $ - 4 $
So, the correct answer is “-4”.

Note: In these types of number problems, we will be given some clues about one or more numbers, and we will use these clues to build an equation. Number problems don’t usually arise on an everyday basis, but they provide a good introduction to practicing the Problem-solving strategy. While solving the number problems, look carefully for the clues words such as and, difference and of.
“Number” word problems are fairly contrived, but they are also fairly standard, so we should learn how to handle them. After all, the point of these problems isn’t their relation to “real life”, but our ability to extract the math from the English.