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?

2 Answers
Oct 8, 2017

See a solution process below:

Explanation:

We can first name our variables. Because they are multiples of 1010 we can write them as: 10m10m and 10n10n.

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

Equation 1: 10m - 10n = 33010m10n=330

Equation 1: 10m + 10n = 71010m+10n=710

Step 1: Solve the first equation for mm:

10m - 10n = 33010m10n=330

10(m - n) = 33010(mn)=330

(10(m - n))/color(red)(10) = 330/color(red)(10)10(mn)10=33010

(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

Oct 8, 2017

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