Question

One number exceeds another by 8. The sum of the numbers is 54. What are the numbers?

Answer 1

The first number is 23 so the second number is 23+8=31`

`23+31=54`

Explanation:

It is difficult to do this without algebra so consider this solution as an introduction to the subject.

By using a letter you are stating that a number exists but you do not know what its value is.

Consider a letter as an empty box into which you may place any value. Sometimes only 1 value in the box (letter) will make the equation/expression work, as in this question.

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`color(green)("Answering your question")`

Let the first unknown number be represented by the letter `n`

Then the second number exceeds the first number by 8 giving `n+8`.

First number is `n`
Second number is `n+8`

The sum of these two numbers is 56

`" "=> n+(n+8)=56`
`color(white)(.)`
`color(white)(.)`

But `n+n` is 2 of `n` which is written as `2n`

`" "color(brown)(=>2n+8=54)`
`color(white)(.)`
`color(white)(.)`

Subtract `color(blue)(8)` from both sides

`" "color(brown)(2n+8color(blue)(-8)=54color(blue)(-8))`

`color(white)(.)`
`color(white)(.)`

But 8-8=0 and 54-8 = 46

`" "2n+0=46" "->" "2n=46`

`color(white)(.)`

Divide both sides by 2

`" "color(brown)((2n)/(color(blue)(2))=46/(color(blue)(2))" "->" "2/(color(blue)(2))xxn=46/(color(blue)(2))`

`color(white)(.)`

But `2/2=1" and "46/2=23`

`" "1xxn=23`

`" "n=23`
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The first number is 23 so the second number is 23+8=31`