Question

Kelly spends 1 4/5 of a hour doing math homework, 3/5 of an hour doing history, and 1 1/2 hours of doing science.a) Kelly spends 1/3 of her time doing math homework working on word problems. What fraction of an hour did she spend on the word problems.b) How many hours did Kelly spend doing homework in total.

Answer 1

Kelly spent

1 4/5 hour doing her math homework

3/5 hour doing history homework

1 1/2 hour doing science homework

a) From the 1 4/5hours she spent doing her math homework, it took her 1/3 of the said time to do a word problem.

What you have to do is calculate how much is 1/3 of 1 4/5hours. To do so you can divide the time by 3:

`1(4)/(5)/3`

Or multiply it by 1/3, which is the same

`1(4)/(5)\cdot(1)/(3)`

I will multiply it by 1/3, to do so, the first step is to express the mixed number as an improper fraction. To express the mixed value as an improper fraction you have to express the whole number as a fraction and add it to the fraction.

So, first, divide the whole number by one to express it as a fraction:

`1(4)/(5)=(1)/(1)+(4)/(5)`

Second, to add them, you have to express both fractions using the same denominator. The common denominator between both fractions is 5, so multiply the numerator and denominator of the first fraction by 5

`(1\cdot5)/(1\cdot5)+(4)/(5)=(5)/(5)+(4)/(5)`

Now you can add both fractions

`(5)/(5)+(4)/(5)=(5+4)/(5)=(9)/(5)`

Next, since the mixed value is expressed as a fraction, you can multiply it by 1/3:

`(9)/(5)\cdot(1)/(3)=(9\cdot1)/(5\cdot3)=(9)/(15)`

The fraction 9/15 is not in its simplest form, to simplify it you have to divide the numerator and denominator by a number that is a common factor for both values. The common factor between 9 and 15 is 3, so you have to divide them by 3:

`(9/3)/(15/3)=(3)/(5)`

So, Kelly spent 3/5 of an hour solving the word problem.

b. You have to calculate how much time Kelly spent doing all her homework, that is, her math homework plus her history homework plus her science homework. To determine this you have to add the time it took her to solve the homework for each subject.

As mentioned before, to be able to add fractions you have to express them using the same common denominator, and in the case of mixed numbers, you have to express them as improper fractions first.

Math time:

We have already expressed this time as an improper fraction for item a, so we know that

`1(4)/(5)=(9)/(5)`

History time:

It is already expressed as a fraction

`(3)/(5)`

Science time:

This time is expressed as a mixed number

`1(1)/(2)`

To express it as an improper fraction, first, divide the whole number by one:

`1(1)/(2)=(1)/(1)+(1)/(2)`

Second, multiply the numerator and denominator of the first fraction by 2 to express it with the same denominator as 1/2:

`(1\cdot2)/(1\cdot2)+(1)/(2)=(2)/(2)+(1)/(2)`

Third, now that both fractions are expressed using the same denominator, you can add them:

`(2)/(2)+(1)/(2)=(2+1)/(2)=(3)/(2)`

So 1 1/2 expressed as an improper fraction is 3/2

Now, what is left is to add the three times to determine how much time did Kelly spend doing her homework:

`(9)/(5)+(3)/(5)+(3)/(2)`

As you can see the first two fractions have the same denominator so we can add them:

`\begin{gathered} (9+3)/(5)+(3)/(2) \\ (12)/(5)+(3)/(2) \end{gathered}`

To add 12/5 and 3/2 you have to express them with the same denominator. A common denominator between "5" and "2" is "10", to express both fractions with denominator 10 you have to multiply the first one by 2 and the second one by 5, both their numerators and denominators:

`(12\cdot2)/(5\cdot2)+(3\cdot5)/(2\cdot5)=(24)/(10)+(15)/(10)`

Next is to add them

`(24)/(10)+(15)/(10)=(24+15)/(10)=(39)/(10)`

She spent 39/10 hours doing her homework, you can express it as a mixed number by dividing 39 by 10:

`39/10=3.9`

This means that she spent 3 whole hours plus 9/10 of an hour doing her homework:

`3(9)/(10)`