Lesson Using fractions to solve word problems

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<H2>Using fractions to solve word problems</H2>

The purpose of this lesson is to present some typical word problems that may be solved using fractions.

<H3>Problem 1. Working together to complete a job</H3>Andrew can cover the roof of a house in &nbsp;3&nbsp; days.
Bill can make this job in &nbsp;6&nbsp; days.
How long will it take to Andrew and Bill to complete the job working together?

<B>Solution</B>

Since Andrew can cover the roof of a house in &nbsp;3&nbsp; days, &nbsp;he is covering &nbsp;{{{1/3}}}&nbsp; of the roof area  per day.

Since Bill can complete this job in &nbsp;6&nbsp; days, &nbsp;he is covering &nbsp;{{{1/6}}}&nbsp; of the roof area  per day.

So, &nbsp;working together, &nbsp;Andrew and Bill can cover &nbsp;{{{1/3 + 1/6}}}&nbsp; of the roof area per day.

Now you need to calculate this sum of two fractions, &nbsp;{{{1/3 + 1/6}}}. &nbsp;It requires standard operations:

&nbsp;&nbsp;&nbsp;&nbsp;- to convert fractions to a common denominator, which is equal to &nbsp;<B>6</B>&nbsp; in this case; 
  &nbsp;&nbsp;&nbsp;&nbsp;- then add numerators; 
  &nbsp;&nbsp;&nbsp;&nbsp;- then reduce the resulting fraction:

{{{1/3 + 1/6}}} = {{{2/6 + 1/6}}} = {{{(2+1)/6}}} = {{{3/6}}} = {{{1/2}}}.

Thus, &nbsp;working together, &nbsp;Andrew and Bill can cover &nbsp;{{{1/2}}}&nbsp; of the roof area per day.

Hence, &nbsp;it will take &nbsp;2&nbsp; days for Andrew and Bill to complete this job working together.

<B>Answer</B>. &nbsp;Working together, &nbsp;Andrew and Bill can complete the job in &nbsp;2&nbsp; days.

<H3>Problem 2. Installing solar panels</H3>One team of workers can install solar panels on the roof of a house in &nbsp;15&nbsp; days by covering the entire roof area.
The second team of workers can complete this job in &nbsp;10&nbsp; days.
How long will it take for two teams to complete the job working together?

<B>Solution</B>

Since the first team can install solar panels on the roof of a house in &nbsp;15&nbsp; days by covering the entire roof area, &nbsp;the team is covering &nbsp;{{{1/15}}}&nbsp; of the roof area  per day.

Since the second team can complete this job in &nbsp;10&nbsp; days, &nbsp;this team is covering &nbsp;{{{1/10}}}&nbsp; of the roof area  per day.

Working together, &nbsp;two teams can cover &nbsp;{{{1/15 + 1/10}}}&nbsp; of the roof area per day.

Now you need to calculate this sum of two fractions, &nbsp;{{{1/15 + 1/10}}}. &nbsp;It requires standard operations:

&nbsp;&nbsp;&nbsp;&nbsp;- to convert fractions to a common denominator, which is equal to &nbsp;<B>30</B>&nbsp; in this case; 
  &nbsp;&nbsp;&nbsp;&nbsp;- then add numerators; 
  &nbsp;&nbsp;&nbsp;&nbsp;- then reduce the resulting fraction:

{{{1/15 + 1/10}}} = {{{2/30 + 3/30}}} = {{{(2+3)/30}}} = {{{5/30}}} = {{{1/6}}}.

Thus, &nbsp;working together, &nbsp;two teams can cover &nbsp;{{{1/6}}}&nbsp; of the roof area by solar panels each day.

Hence, &nbsp;it will take &nbsp;6&nbsp; days for two teams of workers to complete this job working together.

<B>Answer</B>. &nbsp;Working together, &nbsp;two teams of workers can complete the job in &nbsp;6&nbsp; days.

<H3>Problem 3. Filling a reservoir with water</H3>One tube can fill the reservoir with the water in &nbsp;12&nbsp; hours.
The second tube can fill the reservoir with the water in &nbsp;36&nbsp; hours, &nbsp;if works separately.
How long will it take to fill the reservoir, &nbsp;if two tubes work simultaneously?

<B>Solution</B>

Since the first tube can fill the reservoir with the water in &nbsp;12&nbsp; hours, &nbsp;it is filling &nbsp;{{{1/12}}}&nbsp; of the reservoir volume per hour.

Since the second tube can fill the reservoir with the water in &nbsp;36&nbsp; hours, &nbsp;it is filling &nbsp;{{{1/36}}}&nbsp; of the reservoir volume per hour.

Working simultaneously, &nbsp;two tubes fill &nbsp;{{{1/12 + 1/36}}} = {{{3/36 + 1/36}}} = {{{(3+1)/36}}} = {{{4/36}}} = {{{1/9}}}&nbsp; of the reservoir volume per hour.

Hence, &nbsp;the reservoir will be filled in &nbsp;9&nbsp; hours, &nbsp;if both tubes work simultaneously.

<B>Answer</B>. &nbsp;It will take &nbsp;9&nbsp; hours to fill the reservoir, &nbsp;if two tubes work simultaneously.

<H3>Problem 4. Emptying a reservoir</H3>An elephant can drink all the water from a reservoir in &nbsp;4&nbsp; days.
The rhinoceros can drink all the water from the same reservoir in &nbsp;12&nbsp; days, &nbsp;if drinks alone.
How long will it take to empty the reservoir, &nbsp;if the elephant and the rhinoceros drink together?

<B>Solution</B>

Since the elephant can drink all the water in &nbsp;4&nbsp; days, &nbsp;he drinks &nbsp;{{{1/4}}}&nbsp; of the reservoir volume per day.

Since the rhinoceros can drink all the water in &nbsp;12&nbsp; days, &nbsp;he drinks &nbsp;{{{1/12}}}&nbsp; of the reservoir volume per day.

Drinking together, &nbsp;two animals empty &nbsp;{{{1/4 + 1/12}}} = {{{3/12 + 1/12}}} = {{{(3+1)/12}}} = {{{4/12}}} = {{{1/3}}}&nbsp; of the reservoir volume per day.

Hence, &nbsp;the reservoir will be emptied in &nbsp;3&nbsp; days, &nbsp;if both animals drink together.

<B>Answer</B>. &nbsp;The reservoir will be emptied in &nbsp;3&nbsp; days, &nbsp;if the elephant and the rhinoceros drink together.

<H3>Problem 5. Filling a reservoir</H3>The tube can fill a reservoir with the water in &nbsp;3&nbsp; days.
An elephant can drink all the water from the reservoir in &nbsp;4&nbsp; days.
How long will it take to fill the reservoir, &nbsp;if the tube fills it and the elephant drinks the water at the same rates?

<B>Solution</B>

Since the tube can fill the reservoir with the water in &nbsp;3&nbsp; days, &nbsp;it is filling &nbsp;{{{1/3}}}&nbsp; of the reservoir volume per day.

Since the elephant can drink all the water of the full reservoir in &nbsp;4&nbsp; days, &nbsp;he is drinking &nbsp;{{{1/4}}}&nbsp; of the reservoir volume per day.

If the tube is filling &nbsp;{{{1/3}}}&nbsp; of the reservoir volume per day and the elephant is drinking &nbsp;{{{1/4}}}&nbsp; of the reservoir volume per day,

then the net balance is &nbsp;{{{1/3 - 1/4}}}&nbsp; of the reservoir volume per day.

Now you need to calculate this difference of two fractions, &nbsp;{{{1/3 - 1/4}}}. &nbsp;It requires standard operations:

&nbsp;&nbsp;&nbsp;&nbsp;- to convert fractions to a common denominator, &nbsp;which is equal to &nbsp;<B>12</B>&nbsp; in this case; 
  &nbsp;&nbsp;&nbsp;&nbsp;- then subtract numerators; 
  &nbsp;&nbsp;&nbsp;&nbsp;- then reduce the resulting fraction:

{{{1/3 - 1/4}}} = {{{4/12 - 3/12}}} = {{{(4-3)/12}}} = {{{1/12}}}.

So, &nbsp;if the tube is filling &nbsp;{{{1/3}}}&nbsp; of the reservoir volume per day and the elephant is drinking &nbsp;{{{1/4}}}&nbsp; of the reservoir volume per day,

then the volume of the water in the reservoir will increase in &nbsp;{{{1/12}}}&nbsp; part of the total volume at the end of each day.

Hence, &nbsp;the reservoir will be filled in &nbsp;12&nbsp; days.

<B>Answer</B>. &nbsp;The reservoir will be filled in &nbsp;12&nbsp; days.

My other lessons on rate-of-work problems in this site are

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/quadratic/lessons/Using-quadratic-equations-to-solve-word-problems.lesson>Using quadratic equations to solve word problems</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Rate-of-work-word-problems/Solving-rate-of-work-problem-by-reducing-to-a-system-of-linear-equations.lesson>Solving rate of work problem by reducing to a system of linear equations</A>.

My other lessons on fractions in this site are

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/NumericFractions/Using-fractions-to-solve-Travel-problems.lesson>Using fractions to solve Travel problems</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/NumericFractions/Calculations-with-fractions.lesson>Calculations with fractions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/NumericFractions/Advanced-problems-on-ratios.lesson>Advanced problems on ratios</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/NumericFractions/Entertainment-problem-on-fractions.lesson>Entertainment problems on fractions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/NumericFractions/OVERVIEW-of-my-lessons-on-fractions.lesson>OVERVIEW of my lessons on fractions</A>