Ratio of Volumes

EXAMPLE:
The corn flakes of a food company come in two boxes: the regular box and the family value box. For the family value box, the length, width, and height of the box have all been increased by 20%.

By what percentage does the volume of the box increase from the regular box to the value box? Round your answer to the nearest percent.

SOLUTION:
Represent the original volume by V = Lwh and the new volume by V(new) = 1.2L(1.2w)(1.2h) because if the length increases by 20%, the new length will be the old length (L) plus 20% or .2L more. That is where the 1.2L for the new length comes from. Likewise for width and height. You can multiply the 1.2 times 1.2 times 1.2 together to get the new volume is 1.728 times Lwh (the old volume). That is 1 plus .728 times the old volume. The 1 just gives you the old volume. The .728 tells you how much it increased. Change that to a percent and you get 72.8%.

In general if something increases by a certain percent, to find the new amount you multiply by that percent plus 100% to get the new total. For example, if you make $12 per hour and get a 10% raise, that means you will be making 110% of your old wage or 1.1 times $12 or $13.20 per hour.

If the original amount and new amount are both known, you can find what percent something increases or decreases by taking the amount of increase or decrease and dividing by the original number. If you know your wage went from $12 to $13.20 per hour, you can find the percent by calculating $1.20 divided by 12 and changing the decimal you get to a percent. Use this procedure when you know both amounts but not the percent.

In these volume problems you are finding a percent increase, but you do not have numbers for the old and new volumes so you can't do it this way. If they simply said the old volume was 120 cubic cm and the new is 150 cubic cm, then the percent increase would be 30/120 or 25%. Unfortunately, they didn't make it quite that simple.

One more example:

Suppose this time the length is increased by 25% and the width is decreased by 5%. Now, the old volume is still V = Lwh, but the new volume would be (100%+25%)L times (100%-5%)w times h. Notice when we decrease a measurement we subtract from 100%.

Writing these as decimals we get the new volume is 1.25L(.95W)(h) or 1.1875Lwh. This is 1.1875 times the old volume or an increase of .1875 or 18.75%.