Question

You have a piece of wood that is 7 3/4 feet long. You want to cut one piece that is 3 7/12 feet long and one piece that is 4 1/12 feet long. Do you have enough wood? Explain.

Answer 1

Yes, there is enough wood to be cut into the two pieces specified

See explanation below

Explanation:

A good strategy is to convert all the mixed fractions to improper fractions, get the denominators the same and then it becomes easier to handle

`\text{If you have a mixed fraction of the form }a (b)/(c),`
`\text{the equivalent improper fraction is }((a * c + b))/(c)`

`(1) \quad\text{Convert } 7 (3)/(4) \text{ to improper fraction }:\\= (7 * 4 + 3)/(4)\\= (31)/(4)`

`(2)\quad \text{Convert $ 3(3)/(12) $ to improper fraction:}\\\\= (3 * 12 + 3)/(12)\\\\= (43)/(12)`

`(3)\quad \text{Convert $ 4\:(1)/(12) $ to mixed fraction: }\\\\= (4 * 12 + 1)/(12)\\\\= (49)/(12)\\`

Therefore the two pieces of wood are

`\text{$ (43)/(12)$ and $ (49)/(12)$ feet long}}`

Adding up both we get

`(43)/(12)+ (49)/(12)`

Since the denominator is the same, we can simply add the numerators and use the common denominator as the denominator of the result:
43 + 49 = 92

So

`(43)/(12)+ (49)/(12) = (92)/(12)`

We have to compare this to
`(31)/(4)`

To get the same denominator, 12, multiply numerator and denominator by 3

`(31 * 3)/(4* 3) = (93)/(12)`

Compare sum of the lengths of the two pieces of wood
`(92)/(12)` with the length of the original piece
`(93)/(12)`

Since the denominators are the same, just compare the numerators
93 > 92
So length of original piece > sum of lengths of the cut pieces

So there is enough wood to cut into the two pieces specified(just about)

Answer 2

Yes, you have enough wood.

Explanation:

To find out if there is enough wood, we can add the lengths of the two pieces we want to cut:

3 7/12 + 4 1/12 = 7 8/12 feet

We can simplify 7 8/12 by reducing the fraction:

7 8/12 = 7 2/3 feet

So we need 7 2/3 feet of wood to cut the two pieces we want.

Since the piece of wood we have is 7 3/4 feet long, we have enough wood. In fact, we have a little bit extra - if we cut the two pieces we want, there will be 1/12 of a foot (or 1 inch) left over.