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Problem Solving Plan, Diagrams

Develop selected strategies for problem solving.

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Problem Solving Plan, Diagrams
Credit: Andrea Goh
Source: https://www.flickr.com/photos/thebokeeffect/7308563880/
License: CC BY-NC 3.0

Katie is trying out a new cake recipe. The cake recipe said to add \begin{align*}2 \frac{1}{4}\end{align*} cups of flour. The cake turned out to be too moist. She tried the recipe again, but added \begin{align*}\frac{1}{3}\end{align*} cup of flour to the recipe. The cake was much better this time. How many cups of flour did Katie add in all?

In this concept, you will learn how to use the problem solving strategy: draw a diagram.

Problem-Solving with Diagrams

Drawing a diagram or a picture is a strategy to help you solve many different problems involving fractions. The first thing that you should do when approaching a problem is to identify the given information and what the problem is asking for. Here is an example.

John ate \begin{align*}\frac{1}{5}\end{align*} of the cake. What fraction is left?

First, the problem gives you the amount of cake that John ate. The solution is how much cake he has left. This problem requires subtraction. Draw a diagram to show what is known about John and his cake.

This is a diagram of fraction bars to represent John’s cake. The blue section shows how much of the cake John has eaten. The white bars represent the amount of cake that is left. You can see that there are four-fifths left.

The answer to the problem is John has \begin{align*}\frac{4}{5}\end{align*} of the cake left.

You can use a diagram to solve a problem or create a numeric expression. Often times both ways will work, but one will make more sense than the other. Let's look at another problem. 

Shannon jogged \begin{align*}1 \frac{3}{20}\end{align*} miles yesterday. Today, she jogged \begin{align*}\frac{1}{2}\end{align*} mile. How many total miles did Shannon jog?

Method 1 –– Draw a diagram:

One way to solve this problem is to draw a diagram. Look at the first distance that Shannon jogged. Draw two same-sized rectangles. Divide the rectangles into 20 equal-sized sections. Then shade \begin{align*}1 \frac{3}{20}\end{align*} of the diagram.

License: CC BY-NC 3.0

This represents the \begin{align*}1 \frac{3}{20}\end{align*} miles that Shannon jogged yesterday.

Shannon also jogged \begin{align*}\frac{1}{2}\end{align*} mile today.

So, shade \begin{align*}\frac{1}{2}\end{align*} of the partially filled rectangle to represent the distance she jogged today.

License: CC BY-NC 3.0

The diagram is \begin{align*}1 \frac{13}{20}\end{align*} shaded. Therefore, Shannon jogged a total of \begin{align*}1 \frac{13}{20}\end{align*} miles on those two days.

Method 2–– Set up an addition problem:

To find out how many miles she jogged all together, add \begin{align*}1 \frac{3}{20} + \frac{1}{2}\end{align*}. The fractional part of the mixed number has a different denominator than \begin{align*}\frac{1}{2}\end{align*}. Find the least common multiple (LCM) of both denominators. The least common multiple of 20 and 2 is 20. Next, rename the fractions with a common denominator.

\begin{align*}\frac{1}{2} = \frac{10}{20}\end{align*}

Now you can add the fractions together.

\begin{align*}1 \frac{3}{20} + \frac{10}{20} = 1 \frac{13}{20}\end{align*}

Notice that the answer is the same. Both methods will produce the same result. Choose the method that you find easiest when working on problems like this.


Example 1

Earlier, you were given a problem about Katie's cake recipe.

Katie started with \begin{align*}2\frac{1}{4}\end{align*} cups of flour, but added \begin{align*}\frac{1}{3}\end{align*} cup of flour to the recipe. Add the flour quantities to find the total amount of flour used for the cake.

\begin{align*}2\frac{1}{4} + \frac{1}{3}\end{align*}

Look at the fractions. They do not have a common denominator.

First, rewrite the fractions with a common denominator. The LCM of 3 and 4 is 12.

 \begin{align*}2\frac{1}{4} = 2\frac {3}{12} \quad \quad \frac{1}{3} = \frac {4}{12}\end{align*}

\begin{align*}2\frac{1}{4} + \frac{1}{3} =2\frac{3}{12} + \frac{4}{12}\end{align*} 

Then, draw a diagram to find the sum

License: CC BY-NC 3.0

\begin{align*}2\frac{3}{12} + \frac{4}{12} = 2\frac{7}{12}\end{align*}

Katie used a total of \begin{align*}2\frac{7}{12}\end{align*} cups of flour. 

Example 2

Use problem solving strategies-- make a diagram and set up an addition problem-- to solve.

Teri ran \begin{align*}1 \frac{1}{2}\end{align*} miles yesterday, and she ran \begin{align*}2 \frac{1}{2}\end{align*} miles today. How many miles did she run in all?

If John ran 7 miles, what is the difference between his total miles and Teri’s total miles? How many miles have they run altogether?

First, draw a diagram to find the total miles for Teri.

Let's start with Terri.

License: CC BY-NC 3.0

\begin{align*}1 \frac{1}{2} + 2 \frac{1}{2} = 4\end{align*} 

Terri ran 4 miles.

Next, find the difference between Teri and John's total miles. 

License: CC BY-NC 3.0

7 - 4 = 3

There is a difference of 3 miles.

Finally, find the sum of their total miles.

License: CC BY-NC 3.0

7 + 4 = 11

Together, they ran 11 miles. 

Example 3

Draw a diagram to solve the problem: \begin{align*}2 \frac{1}{3} + 4\end{align*}.

License: CC BY-NC 3.0

The sum is \begin{align*}6 \frac{1}{3}\end{align*}.

Example 4

Draw a diagram to solve the problem: \begin{align*}\frac{4}{5} - \frac{1}{5}\end{align*}.

License: CC BY-NC 3.0

The difference is \begin{align*}\frac{3}{5}\end{align*}.

Example 5

Draw a diagram to solve the problem: \begin{align*}\frac{3}{4} + \frac{2}{4}\end{align*}.

License: CC BY-NC 3.0

The sum is \begin{align*}1 \frac{1}{4}\end{align*}.


Solve each of the following problems by using a problem solving strategy.

  1. Tyler has eaten one-fifth of the pizza. If he eats another two-fifths of the pizza, what part of the pizza does he have left?
  2. What part has he eaten in all?
  3. How many parts of this pizza make a whole?
  4. Maria decides to join Tyler in eating pizza. She orders a vegetarian pizza with six slices. If she eats two slices of pizza, what fraction has she eaten?
  5. What fraction does she have left?
  6. If Tyler was to eat half of Maria’s pizza, how many pieces would that be?
  7. If Maria eats one-third, and Tyler eats half, what fraction of the pizza is left?
  8. How much of the pizza have they eaten altogether?
  9. John and Terri each ran 18 miles. If Kyle ran half the distance that both John and Teri ran, how many miles did he run?
  10. If Jeff ran \begin{align*}3 \frac{1}{2}\end{align*} miles, how much did he and Kyle run altogether?
  11. What is the distance between Jeff and Kyle’s combined mileage and John and Teri’s combined mileage?
  12. Sarah gave Joey one-third of the pie. Kara gave him one-fourth of another pie. How much pie did Joey receive altogether?
  13. Is this less than or more than one-half of a pie?
  14. Who gave Joey a larger part of the pie, Kara or Sarah?
  15. What is the difference between the two fractions of pie?

Review (Answers)

To see the Review answers, open this PDF file and look for section 6.18. 


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Diagram A diagram is a drawing used to represent a mathematical problem.
Problem Solving Problem solving is using key words and operations to solve mathematical dilemmas written in verbal language.

Image Attributions

  1. [1]^ Credit: Andrea Goh; Source: https://www.flickr.com/photos/thebokeeffect/7308563880/; License: CC BY-NC 3.0
  2. [2]^ License: CC BY-NC 3.0
  3. [3]^ License: CC BY-NC 3.0
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  5. [5]^ License: CC BY-NC 3.0
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  10. [10]^ License: CC BY-NC 3.0

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