find the numbers.

Question

One number is 8 times a first number. A third number is 100 more than the first number. If the sum of the three numbers is 740​,

Answer 1

With these word problems, they're looking for you to find the equations within the sentences/paragraph they provide. So when reading these, try to imagine the equations they're hinting at. For instance, the first sentence ("One number is 8 times a first number") is the same thing as saying B = 8*A (with B representing the missing "one number" and A representing the "first number".

Let's do this for all three equations they're hinting at. We have the first equation (B=8A). For the 2nd sentence ("Third number is 100 more than the first number"), this is saying that if you add 100 to the first number, you get the third number. So, with C as the variable representing this third number, we have the equation A + 100 = C.

For the third sentence ("Sum of the numbers is 740"), that just says to add all three together and you get 740. Thus, A + B + C = 740.

You have everything you need to solve this problem now! The way you would do this is by plugging in variables into other equations. To do this, you need to have one of the equation in such a format that a variable is isolated on one side of the equal sign and the rest of the equation is on the right.

(Brief aside: For instance, using a different example: if we have two equations - one that says that X + Y = 20 and X = 12, you could just plug in X into the first equation to find that Y = 8. You can do the same thing if you have a second equation that says X = Y + 4. Even though this second version of X doesn't provide an exact number like 12, you can still do the same thing you did before - plug this value into the first equation. Instead of X + Y = 20, you would have (Y + 4) + Y = 20 (because X = Y + 4). Then, you could also solve for Y by combining like terms to get 2Y + 4 = 20, then you can again solve for Y = 8. We're doing the same thing, only we don't know what that 2nd variable is supposed to be yet. We'll solve for it though!)

Our three equations so far:

B = 8A

A + 100 = C

A + B + C = 740

We know that B = 8A and that C = A + 100. Let's plug both of these into the third equation.

A + B + C = 740

A + (8A) + (A + 100) = 740

Now just solve for A. First, combine like terms:

10A + 100 = 740

Isolate 10A by subtracting 100 from both sides. This cancels out the 100 on the left side while keeping the two sides equivalent.

10A = 640

Finally, isolate A by dividing both sides by 10 (as (10*A)/10 is the same as 1*A) to get the value of A

A = 64

Now that you have A, you can solve for the other values by plugging A into those other two equations. Finally, make sure the math worked out by plugging all three values back into the first equation!

Good luck!

Answer 2

Let "x" be "the first number"

That means one number is 8x.

It also means the third number is x + 100.

Their sum would be (x) + (8x) + (x + 100) and since the sum is 740 then:

(x) + (8x) + (x + 100) = 740

x + 8x + x + 100 = 740

10x + 100 = 740

10x = 640

x = 64

So the first number is 64, another number is 8•64 or 512 and the third number is 64 + 100 or 164