Add and subtract fractions with unlike denominators: word problems

Question

Gary's chemistry textbook weighs 5/6 of a pound and his geometry textbook weighs 1/12 of a pound. How much more does the chemistry textbook weigh than the geometry textbook

Answer 1

When trying to add or subtract fractions with unlike denominators, our goal more or less is to make their denominators the same.

The reason why we need to do this is because of how fractions work. In each fraction, the numerator represents the number of equal parts that a value has relative to a given whole, or the denominator. If I eat one half (1/2) of a pizza, to take a classic example, what I have done is divided that pizza into two equal parts (the denominator) and eaten one of them (the numerator). If I ate one fourth (1/4) then I would have divided the pizza into four equal parts and eaten one, leaving three equivalently sized slices.

The problem is that when our denominators are different, the meaning of the "parts" represented in the numerator changes. In both of the above examples I ate "one part" of the pizza (the numerator of how much I ate was 1), but the part eaten in the first example was much larger than the part of the second – a half versus a fourth. Comparing these numerators directly is thus very difficult because the actual quantity (of pizza) they entail is not the same.

In contrast, if we recognized the fact that (1/2) is equal to (2/4), we would understand that in my first example I ate what would be equivalent to two slices in the second example (when I ate 1/4th). Noticed how this involves changing the denominator of my first example to match the denominator of the second (so that both equaled 4). This shows that when the denominators of two fractions match, the values of their numerators are measured on the same scale, allowing us to relate them by addition or subtraction (for instance, if someone ate 1/2 of a pizza, and then another person ate 1/4th, the above would allow us to add how much they ate together: 3/4ths of the pizza, or 2/4 + 1/4).

Returning to the problem at hand: the easiest way we can make the denominators of these fractions the same is to find what is called their lowest common denominator. The lowest common denominator (or LCD) is the smallest multiple that is shared between the numbers of two (or more) denominators, or the lowest number that both denominators could be simplified into (without having a fraction in the numerator). Why the smallest? Because the larger our numbers, the harder they are to calculate with, so keeping our denominator small is to our advantage.

In this case, 12 is a multiple of 6, and since six 6 is not a multiple of 12 the 12th of Gary's geometry textbook is our goal. To make the denominator in the weight of his chemistry textbook 12, we need to multiply it and the numerator by 2/2:

5 2 10

– * – = –

6 2 12

** Notice how 2/2 is equal to 1! This means that while we are doubling both the numerator and denominator in our original fraction, what we are actually doing multiplying our original fraction by 1 which will always equal itself! This is an important point, because we don't want the weight of Gary's chemistry textbook to change; we just want it to be measured in the same units as his geometry textbook! **

What we have done by changing the denominator in this way is we have divided 1 pound into 12 equal parts and found how many of those parts each textbook weighs. This was done for us in the case of Gary's geometry textbook, but now we know that the 5 parts of 6 of his chemistry textbook is the same as 10 parts of the same 12 used to measure his geometry textbook

Gary's chemistry textbook weighs the equivalent of 10 one twelfths of a pound, whereas his geometry textbook ways only 1 one twelfth of a pound. Since we are looking for the difference between these weights we just need to find how many twelfths of a pound more does Gary's chemistry textbook weigh than his geometry textbook, which we can do by subtracting them:

10 1 9

–– - –– = ––

12 12 12

And there you have it, Gary's chemistry textbook weighs 9 twelfths of a pound more than his geometry textbook (9/12) or, if you simplify the fraction by dividing by 3/3 (using the same principle as above!): Gary's chemistry textbook weighs 3/4 pounds more than his geometry textbook.

Answer 2

(5 / 6) - (1 / 12)

(10 / 12) - (1 / 12)

Cank you complete and answer?