The relationship between a given set of values is said to be proportional when their ratio is constant. This discussion will be on graphing proportional relationships and finding out their characteristics.

## What are Graphing Proportional Relationships?

- A graphing proportional relationship between two variables is a relationship where the ratio between the two variables is always the same. For example, we consider the relationship between x and y. When x is one, y is three. When x is two, y is six. When x is 9, y is 27. This is a proportional relationship. Why is that? Because the ratio between x and y is always the same value. The ratio between y and x is also constant.

x | y |

1 | 3 |

2 | 6 |

9 | 27 |

yx = 31 =62 = 279 = 3

**Let us look at this set of values.**

A | b |

1 | 3 |

2 | 6 |

10 | 35 |

When a is 1, b is 3. When a is 2, b is 6, but when a is 10, b is 35.

Ratio ba = 31= 62 not equal to 3510 . So, this is not a proportional relationship. The key to identifying proportional relationships is to look at the different values the variables take on. Notice when one variable is one value, then what does the other variable become and then take the ratio between them. In the above example, we took the ratio y to x and found the y to x ratio or y/x. The ratio of y/x was always the same, so it was a proportional relationship. Ratios and fractions are interconvertible. We can turn a ratio into a fraction and vice versa.

## Which Graph Represents a Proportional Relationship?

**Characteristics**

**Graphing proportional relationships has the following features:**

- The graph of all the proportional relationships is a straight line
- It passes through the origin (0,0)

**Let us plot a few graphs with proportional relationships**

**Q1: Form a graph on the following data with proportional relationships.**

Hours | Miles walked |

1 | 2 |

2 | 4 |

3 | 6 |

4 | 8 |

5 | 10 |

**Ans: **

We will graph hours against the x (horizontal) axis and miles walked against the y (vertical) axis.

**Q2: Construct a graph of the following relationships to determine if they are proportional.**

T-shirts sold | Fundraiser money kept |

2 | $1.00 |

3 | $1.50 |

4 | $2.00 |

5 | $2.50 |

6 | $3.00 |

**Ans:**

We will graph money made against the y-axis and t-shirts sold against the x-axis. If we sell two t-shirts, we raise $1, and if we sell three t-shirts, we raise 1.5 dollars. Similarly, if we sell four t-shirts, we raise 2 dollars, and if we sell five t-shirts, we raise $2.5.

When we represent these values in a graph, we notice that it is a straight line. It also passes through the origin (0,0). Both the requirements of a proportional relationship have been fulfilled. So, it is a proportional relationship.

**Q3: Draw a graph of the following set of values.**

Arrows Kaz Fires | Bulls Eyes that Kaz gets |

0 | 0 |

1 | 1 |

2 | 2 |

10 | 10 |

**Ans: **We plot arrows fired against the x-axis and bullseye against the y-axis.

- We obtain a graph with a straight line.
- Does it go through the origin? Yes, it does.
- So, it is a graph of proportional relationships

### Graphing Non-proportional Relationships

Let us now solve a few problems where we need to plot graphing non-proportional relationships. Differentiating between the two will help us understand the concept better.

**Q1: Plot a graph of the following relationships.**

Rounds played | Rounds on par |

3 | 2 |

6 | 3 |

9 | 4 |

12 | 5 |

**Ans:** We plot rounds played against the x-axis and the rounds on par against the y-axis.

After plotting the graph, we need to check a couple of things.

- Is it a straight line? Well, yes! It looks like a straight line.
- Does the line start from the origin? No!
- So, the given set of values does not have a proportional relationship. This is because the graph of values with a proportional relationship fulfills two conditions— it is a straight line that passes through the origin. In this scenario, only one of the conditions is being fulfilled.

**Q2: Plot a graph of the following data.**

Time Studying | Correct Answers |

0 | 0 |

1 | 1 |

2 | 4 |

3 | 9 |

**Ans:** We plot time studying along the x-axis and correct answers along the y-axis.

- When we graph the given value, we notice that we get a curved line.
- The line passes through the origin.
- Both the conditions of proportional relationships are not being fulfilled. So, the given set of values doesn’t share a proportional relationship.

### Problems Involving Ratios and Proportions:

**An island has 540 cats and 675 dogs. What is the ratio of cats to dogs on this island?**

**Ans:** Since cats come first in the question, we will put it on top of the fraction. Dogs came second, so we put their number on the bottom of the fraction.

catsdogs = 540675

Now, we will simplify this fraction. Because the last digits of the two values are 0 and 5, both values are divisible by 5.

540675 5 = 108135

We need to simplify it further if possible. Both 108 and 135 are divisible by 9.

108135divided by 9 =1215 . We can reduce it further by dividing the values by 3. So, we get 45. So, the ratio of cats to dogs is 45.

**The ratio of boys to girls in a class is 87. If there are 40 boys, how many girls are there in the class?**

**Ans:** The ratio of boys to girls is 87. If there are eight boys in the class, there will be seven girls in that same class.

Boys: Girls

As the question is asking to find out the number of girls in the class when there are 40 boys,

8:40 = 7:x

Let us cross multiply the equation

8x = 40 x 7

x = (40/8) x 7 (It is easier to write 40 as 87)

X = (5)7 (8 and 8 get cancelled)

X = 35

So, if there are 40 boys in a class, there will be 35 girls in the same class. We can think about it conceptually. Let us say we have a ratio of 8:7, and we want to expand it to 40:x. To go from 8 to 40, we need to multiply eight by five. So, to keep the ratio the same, we have to multiply seven by five too. It turns out to be 35. That is a quick way to get the answer mentally.

**Eve can make 14 cakes in 6 hours. How many cakes can she make in 15 hours?**

**Ans:** To solve this problem, let us put up a proportion. We are dealing with two things— the number of cakes she can make and the time she can make them. So, we put two fractions separated by an equal sign. We put the information associated with time on the right side and the information associated with cakes on the left side. Since 14 cakes correspond to 6 hrs, we put these two values on the top of the fraction.

14:x = 6 hours:15 hours

Now let us cross multiply

6 x = 14 x15

6x = 210

x = 210/6

X = 35

So, in 15 hours, Eve can make 35 cakes.

Now, let us try solving this problem the fast way.

The ratio of cakes to hours is initially 14:6. We want to know how many cakes Eve can make in 15 hours. What do we need to multiply by six to get 15? It is 2.5. So, we multiply six by 2.5 to get 15. Similarly, to find x, we need to multiply 14 by 2.5. The answer is 35, which is the same as the above method.

### What is a Proportional Relationship on a Graph – Applications in Real Life

- When we fill up our car fuel tank at a petrol pump, there is a proportional relationship between the gallons of fuel we put in and the money we need to pay.
- If one pack of chips costs us 10$, two packs will cost 20$, and three packs will cost 30$. There is a proportional relationship between the packs of chips we buy and the dollars we pay.
- If someone gets 100$ pay for working for one hour, he will get 200$ for two hours’ work and 300$ for three hours’ work. There is a proportional relationship between the number of hours he works and the amount he gets paid.
- If someone is on a weight loss journey and loses 1kg in 10 days, 2kgs in 20 days, and 3 kg in 30 days. There is a proportional relationship between the amount of weight he is losing and the days taken to lose that weight. This relationship will help decide the amount of weight he will lose over the next months.

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