Question

The difference between two multiples of ten is 330. The total of them is 710. What are the two numbers?

is there a way to do this calculation without using simultaneous equations?

Answer 1

See a solution process below:

Explanation:

We can first name our variables. Because they are multiples of `10` we can write them as: `10m` and `10n`.

From the information in the problem we can write two equations:

Equation 1: `10m - 10n = 330`

Equation 1: `10m + 10n = 710`

Step 1: Solve the first equation for `m`:

`10m - 10n = 330`

`10(m - n) = 330`

`(10(m - n))/color(red)(10) = 330/color(red)(10)`

`(color(red)(cancel(color(black)(10)))(m - n))/cancel(color(red)(10)) = 33`

`m - n = 33`

`m - n + color(red)(n) = 33 + color(red)(n)`

`m - 0 = 33 + n`

`m = 33 + n`

Step 2: Substitute `(33 + n)` for `m` in the second equation and solve for `n`:

`10m + 10n = 710` becomes:

`10(33 + n) + 10n = 710`

`10[(33 + n) + n] = 710`

`(10[33 + n + n])/(color(red)(10)) = 710/(color(red)(10))`

`(color(red)(cancel(color(black)(10)))[33 + 2n])/cancel(color(red)(10)) = 71`

`33 + 2n = 71`

`33 - color(red)(33) + 2n = 71 - color(red)(33)`

`0 + 2n = 38`

`2n = 38`

`(2n)/color(red)(2) = 38/color(red)(2)`

`(color(red)(cancel(color(black)(2)))n)/cancel(color(red)(2)) = 19`

`n = 19`

Step 3: Substitute `19` for `n` in the solution to the first equation at the end of Step 1 and calculate `m`:

`m = 33 + n` becomes:

`m = 33 + 19`

`m = 52`

Solution:

If `m = 52` then `10m = 10 xx 52 = 520`

If `n = 19` then `10n = 10 xx 19 = 190`

`520 - 190 = 330`

`520 + 190 = 710`

The numbers are: 520 and 190

Answer 2

the first number is `520` and the second one is `190`

Explanation:

Let the numbers be `10x` and `10y`

Then the difference is:

`10x-10y=330`

and their sum is:

`10x+10y=710`

If we sum the equations side by side, we'll get:

`10xcancel(-10y)+10xcancel(+10y)=330+710`

that's

`20x=1040`

and `x=52`

We can find y by subtracting the equations side by side:

`cancel(10x)-10ycancel(-10x)-10y=330-710`

`-20y=-380`

that's

`y=19`

then the first number is `10x=10*52=520`

and the second one is `10y=10*19=190`