1. ## consecutive integers help

find three consecutive integers such that the sum of the first and the third is 80?

the difference of two numbers is 46.the larger of the two numbers is 3 times the smaller. find the value of the two numbers.

2. Originally Posted by BeBeMala
find three consecutive integers such that the sum of the first and the third is 80?

let x = the first integer ...

x + (x+2) = 80

the difference of two numbers is 46.the larger of the two numbers is 3 times the smaller. find the value of the two numbers.

x - y = 46

x = 3y
...

3. Say your two consecutive numbers are :

$\displaystyle x$, $\displaystyle x + 1$ and $\displaystyle x + 2$

The sum of the first and the third is $\displaystyle 80$, so $\displaystyle x + x + 2 = 80$, which simplifies to $\displaystyle 2x + 2 = 80$, or $\displaystyle 2x = 78$, and finally $\displaystyle x = 39$. Substitute back this value into your consecutive integers. Your consecutive integers are :

$\displaystyle 39$, $\displaystyle 39 + 1$ and $\displaystyle 39 + 2$, that is, your integers are $\displaystyle 39$, $\displaystyle 40$ and $\displaystyle 41$.

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Say the two numbers are $\displaystyle x$ and $\displaystyle y$, such as $\displaystyle x > y$.

- Their difference is $\displaystyle 46$, so $\displaystyle x - y = 46$.

- The larger of the two numbers ($\displaystyle x$) is $\displaystyle 3$ times the smaller ($\displaystyle y$), so $\displaystyle x = 3y$.

You have $\displaystyle x - y = 46$, and $\displaystyle y = 3x$. Substitute the second into the first :

$\displaystyle (3y) - y = 46$

Which simplifies to :

$\displaystyle 2y = 46$

Thus $\displaystyle y = 23$.

Plug this back into the first equation to find $\displaystyle x$ :

$\displaystyle x - (23) = 46$, so $\displaystyle x = 46 + 23 = 69$.

Your two numbers are $\displaystyle 23$ and $\displaystyle 69$.