how do I solve problems involving sums of quantities?

Question

if 3 is added to a number and this sum is doubled, the result is 2 more than the number.  Find number

Answer 1

Hi, Vanessa.

Math is sometimes very much like learning another language. You have to "translate" the words into a math equation.

You can pick any letter to stand for an unknown number, but most often we use n or x.

So, starting at the beginning of your question, we have:

"three added to a number"          n + 3

"this sum is doubled"                 (n + 3) times 2

We usually write it with the single digit first      2(n+3)

"the result"     is our   2(n + 3)

"is"                    =

"2 more than the number"         n + 2

So,            2(n + 3) = n + 2

There is your equation. To solve it, you have to use the distributive property to multiply 2 by both terms in the parentheses:

2 * n + 2 * 3 = n + 2       which is:

2n + 6 = n + 2

You have the variable on both sides of the equation, so you subtract it from both sides (remember that you have to do the same thing to both sides of the equation to keep it in balance).

2n - n + 6 = n - n + 2

2n - n   is really    2n - 1n,  so the left side of the equation is   1n + 6  (and we can write is as  without the 1); on the right side, the  n - n cancels out.

n + 6 = 2

We then subtract 6 from both sides:

n + 6 - 6 = 2 - 6   and we get:

n = -4

You can now read the original sentence and plug  -4 in:

3 + -4 = -1;  doubled is -2;  the result is 2 more than  -4.   -2 is 2 more than -4, so it works. :-)

Hope this helps!

Kathye

Answer 2

if 3 is added to a number and this sum is doubled, the result is 2 more than the number. Find number

This is a classic form of algebra word problem. We need to pick a variable name to represent the number we are trying to find. I like N for number. Let N be the number we are trying to find.

This is how I decode the word problem to make an equation:

if 3 is added to a number...

3 + N

...and this sum is doubled...

2(3 + N)

...the result is...

2(3 + N) =

...2 more than the number

2(3 + N) = 2 + N

The word “is” usually means an equal sign. The rest of the problem describes the sides of the equation.

Now we solve the equation.

2(3 + N) = 2 + N

Distribute the 2 over the terms in parentheses.

6 + 2N = 2 + N

We want N terms by themselves and numeric terms on the other side of the equal sign. Subtract N from both sides of the equation. Remember, you can add or subtract at will, as long as you do it to both sides of the equation.

6 + 2N - N = 2 + N – N

6 + N = 2

Now we subtract 6 from both sides.

6 + N – 6 = 2 – 6

N = - 4

This is our answer. The number is – 4.

Looks good, but we should always check our work by plugging the answer into the original equation and seeing if it works.

2(3 + N) = 2 + N

2(3 - 4) = 2 + - 4

2(- 1) = - 2

- 2 = - 2 CHECK.