Relationships are not restricted to humans and animals alone. In mathematics, some relationships exist between numbers and quantities.

In this article, we will be looking at the multiplicative relationship between quantities.

## Multiplicative relationship definition

A multiplicative relationship between quantities is the relationship that exists when the quantities are directly proportional to each other or they are multiples of each other.

In a multiplicative relationship, there is a constant number that is used to multiply another number to get a corresponding value.

## Multiplicative relationship equation

A multiplicative relationship between two quantities is in the form,

$y=ax$

where $y$ is the output or dependent quantity,

$x$ is the input or independent quantity,

$a$ is the constant, or the constant of multiplication.

The constant $a$ is what determines if the relationship is multiplicative or not. Hence if the product of constant $a$and$x$ will not give $y$, for every input and output, then it is not a multiplicative relationship.

If you have a set of inputs and outputs and you want to determine if their relationship is multiplicative, you take note of the constant and make sure it is the same all through.

We note that multiplication and division operations are related. Hence, if you have the values of $x$ and $y$, you can get $a$ by dividing $y$ into $x$ while making sure that we are not dividing by zero.

$y=ax\phantom{\rule{0ex}{0ex}}a=\frac{y}{x}$

## Multiplicative relationship examples

Let's take some examples to understand what a multiplicative relationship is.

Find the constants of the multiplicative relationships below.

- 4 and 16
- 2 and 6

**Solution**

a. 4 and 16

4 is the input and 16 is the output. You can write this in the form of the multiplicative relationship equation.

$y=ax\phantom{\rule{0ex}{0ex}}16=4a$

Dividing both sides of the equation by 4 to isolate $a$ we get, $\frac{16}{4}=\frac{4a}{4}\phantom{\rule{0ex}{0ex}}a=\frac{16}{4}=4\phantom{\rule{0ex}{0ex}}$

The constant is 4.

b. 2 and 6

2 is the input and 6 is the output. You can write this in form of the multiplicative relationship equation to get the constant.

$y=ax\phantom{\rule{0ex}{0ex}}6=2a$

Dividing both sides of the equation by 2 to isolate $a$ we get,

$\frac{6}{2}=\frac{2a}{2}\phantom{\rule{0ex}{0ex}}a=\frac{6}{2}=3$

The constant is 3.

Let's take another example.

Determine which set in the table below has a multiplicative relationship.

Set A | Set B | ||
---|---|---|---|

Input (x) | Output (y) | Input (x) | Output (y) |

6 | 12 | 3 | 12 |

8 | 13 | 5 | 20 |

11 | 16 | 7 | 28 |

2 | 7 | 15 | 60 |

**Solution**The table above shows two sets A and B that contains pairs of numbers to compare. We want to know which set has a multiplicative relationship and which one does not.Recall the equation for a multiplicative relationship,$y=ax$

Let's see what we will have if the numbers in set A are substituted in our equation.

The first pair of numbers are 6 and 12. Our equation suggests that 6 multiplied by a number will give 12. Substituting in the equation gives

$12=6a\phantom{\rule{0ex}{0ex}}$

Let's simplify further to get the constant $a$,

$12=6a\phantom{\rule{0ex}{0ex}}\frac{12}{6}=\frac{6a}{6}\phantom{\rule{0ex}{0ex}}a=2$

We can't conclude now that set A has a multiplicative relationship because there are still other pairs of numbers to consider and the constant $a$ has to be the same for all of them.

The next pair of numbers are 8 and 13. 13 is the output $y$ and 8 is the input $x$. Let's substitute in the formula to see what constant $a$ would be.

$y=ax\phantom{\rule{0ex}{0ex}}13=8a$

Divide both sides by 8 to isolate $a$, we get

$\frac{13}{8}=\frac{8a}{8}\phantom{\rule{0ex}{0ex}}a=1.63$

The constant we got here is different from the one we got before which means that the numbers in Set A do not have a multiplicative relationship.

Let's consider now Set B.The first pair of numbers are 3 and 12. Let's substitute in our equation to see what our constant would be. $y=ax\phantom{\rule{0ex}{0ex}}12=3a$Divide both sides by 3 to isolate $a$ to get,

$\frac{12}{3}=\frac{3a}{3}\phantom{\rule{0ex}{0ex}}a=4$

The constant $a$ is 4. Let's see if it will be the same for the other numbers in the set.

The next pair of numbers are 5 and 20. Substituting in the equation will give,

$y=ax\phantom{\rule{0ex}{0ex}}20=5a\phantom{\rule{0ex}{0ex}}$

Dividing both sides of the equation by 5, to isolate $a$, we get$\frac{20}{5}=\frac{5a}{5}\phantom{\rule{0ex}{0ex}}a=4$The constant here is 4.

The third pair is 7 and 28. Substituting in the equation will give,$y=ax\phantom{\rule{0ex}{0ex}}28=7a$Dividing both sides by 7, to isolate $a$, we get$\frac{28}{7}=\frac{7a}{7}\phantom{\rule{0ex}{0ex}}a=4$Again, we have 4 as the constant.The last pair of numbers are 15 and 60. Substituting in the equation will give,$y=ax\phantom{\rule{0ex}{0ex}}60=15a\phantom{\rule{0ex}{0ex}}$Dividing both sides by 15, to isolate $a$, we get$\frac{60}{15}=\frac{15a}{15}\phantom{\rule{0ex}{0ex}}a=4$You can see that the constant is 4 all through. This means that Set B has a multiplicative relationship.The multiplicative relationship is not restricted to a particular pair of numbers. **All** numbers when paired together have a multiplicative relationship.

Consider the numbers 12 and 13, where 12 is the input and 13 is the output. If you are asked to find the multiplicative relationship between these numbers, you might be tempted to say that there is none, because there is no whole number that you can use to multiply 12 to give 13.

This is where ratios and fractions come in. The constant in a multiplicative relationship can be expressed in form of a ratio or fraction.

So, yes! 12 and 13 have a multiplicative relationship. If you multiply 12 by $\frac{13}{12}$, you will get 13.

$12\times \frac{13}{12}=13$

When you come across these kinds of situations, you make sure that the denominator of the fraction you want to use to multiply has the same value as the input and the numerator must be the result you desire to get. Hence, the denominator in the fraction above is 12 and the numerator is 13.

Let's take some examples.

Show the multiplicative relationship between the pairs below.

- 15 and 17
- 35 and 27

**Solution**

a. 15 and 17

Here, 15 is the input, and 17 is the output. To show the multiplicative relationship between 15 and 17, we have to multiply 15 by a fraction. The numerator of the fraction must be 17 and the denominator must be 15 so we can cancel out to get 17 as the answer.

Let's recall the multiplicative relationship equation.

$y=ax$

where $y$ is 17

$x$ is 15

$a$ is the constant,

$y=15a\phantom{\rule{0ex}{0ex}}y=15\times \frac{17}{15}\phantom{\rule{0ex}{0ex}}y=\overline{)15}\times \frac{17}{\overline{)15}}\phantom{\rule{0ex}{0ex}}y=17$

$y\phantom{\rule{0ex}{0ex}}$ is 17 as desired and $a$ is $\frac{17}{15}$.

b. 35 and 27

Here, 35 is the input, and 27 is the output. To show the multiplicative relationship between 25 and 27, we have to multiply 35 by a fraction. The numerator of the fraction must be 27 and the denominator must be 35 so we can cancel out to get 27 as the answer.

Recall, the multiplicative relationship equation.

$y=ax$

where $y$ is 27

$x$ is 35

$a$ is the constant

$y=35a\phantom{\rule{0ex}{0ex}}y=35\times \frac{27}{35}\phantom{\rule{0ex}{0ex}}y=\overline{)35}\times \frac{27}{\overline{)35}}\phantom{\rule{0ex}{0ex}}y=27$

$yis27$ as desired and $ais\frac{27}{35}$.

Let's take another example.

Determine which set in the table has a multiplicative relationship.

Set A | Set B | ||
---|---|---|---|

Input (x) | Output (y) | Input (x) | Output (y) |

3 | 7 | 5 | 2 |

4 | 5 | 10 | 4 |

10 | 2 | 20 | 8 |

7 | 9 | 15 | 6 |

**Solution**

We want to know which set of numbers have a multiplicative relationship. Let's start with Set A.

The first pair of numbers in set A are 3 and 7 where 3 is the input and 7 is the output.

Recall the multiplicative relationship equation,

$y=ax$

We will substitute in the equation to get the constant,

$7=3a\phantom{\rule{0ex}{0ex}}\frac{7}{3}=\frac{3a}{3}\phantom{\rule{0ex}{0ex}}a=\frac{7}{3}$

The constant is $\frac{7}{3}$.

Let's find out if the next pair will give the same constant.

The next pair is 4 and 5 where 4 is the input and 5 is the output.

Let's substitute in the multiplicative relationship formula.

$y=ax\phantom{\rule{0ex}{0ex}}5=4a\phantom{\rule{0ex}{0ex}}\frac{5}{4}=\frac{4a}{4}\phantom{\rule{0ex}{0ex}}a=\frac{5}{4}$

The constant here is different from the first. This means that the numbers in set A do not have a multiplicative relationship.

Let's take a look at set B.

The first pair of numbers are 5 and 2 where 5 is the input and 2 is the output.

Let's substitute in the multiplicative relationship formula.

$y=ax\phantom{\rule{0ex}{0ex}}2=5a\phantom{\rule{0ex}{0ex}}\frac{2}{5}=\frac{5a}{5}\phantom{\rule{0ex}{0ex}}a=\frac{2}{5}$

The constant is $\frac{2}{5}$. If the rest of the pair have $\frac{2}{5}$ as their constant, then there is a multiplicative relationship.

The next pair is 10 and 4 where 10 is the input and 4 is the output.

Let's substitute in the multiplicative relationship formula.

$y=ax\phantom{\rule{0ex}{0ex}}4=10a\phantom{\rule{0ex}{0ex}}\frac{4}{10}=\frac{10a}{10}\phantom{\rule{0ex}{0ex}}a=\frac{4}{10}=\frac{2}{5}$

We have a constant of $\frac{2}{5}$ which is the same as the first. Let's try the last two pair.

The third pair is 20 and 8 where 20 is the input and 8 is the output.

Substituting in the formula will give,

$y=ax\phantom{\rule{0ex}{0ex}}8=20a\phantom{\rule{0ex}{0ex}}\frac{8}{20}=\frac{20a}{20}\phantom{\rule{0ex}{0ex}}a=\frac{8}{20}=\frac{2}{5}$

The constant is $\frac{2}{5}$ again.

The last pair is 15 and 6 where 15 is the input and 6 is the output.

Let's substitute in the multiplicative relationship formula.

$y=ax\phantom{\rule{0ex}{0ex}}6=15a\phantom{\rule{0ex}{0ex}}\frac{6}{15}=\frac{15a}{15}\phantom{\rule{0ex}{0ex}}a=\frac{6}{15}=\frac{2}{5}$

Again, we have the constant to be $\frac{2}{5}$. This means that the numbers in set B have a multiplicative relationship.

## Multiplicative relationship graph

Multiplicative relationships can also be represented in a graph. You can use the multiplicative relationship equation to derive pairs of numbers that can be plotted on a graph.

A graph representing multiplicative relationships is a **straight line** and it always starts from the origin, because anything multiplied by zero is zero.

Let's see some examples.

If $y=4x$ complete the table below and plot the graph.

x | y |

0 | |

1 | |

2 | |

3 |

**Solution**

The equation given is $y=4x$ and you will notice that it is in the form of the multiplicative relationship equation

$y=ax$

What we need to do is substitute the values of $x$ in the equation given to get $y$.

When $x=0$,

$y=4\times 0\phantom{\rule{0ex}{0ex}}y=0$

When $x=1$,

$y=4\times 1\phantom{\rule{0ex}{0ex}}y=4$

When $x=2$,

$y=4\times 2\phantom{\rule{0ex}{0ex}}y=8$

When $x=3$,

$y=4\times 3\phantom{\rule{0ex}{0ex}}y=12$

Now let's put the results we've gotten in the table.

x | y |

0 | 0 |

1 | 4 |

2 | 8 |

3 | 12 |

Now that we've completed the table, let's plot the graph.

We can see that the graph has the characteristics of a multiplicative relationship graph which is that it is linear and it starts from the origin (0, 0).

So far, we've shown what multiplicative relationships are and how to solve them using numbers. Although, the numbers represent values of quantities, let's now look at a real-world situation concerning multiplicative relationships.

Let's take some examples.

A company pays its workers £13 per hour. Show the multiplicative relationship here, create a table and plot a graph from the information on the table.

**Solution**

The question says a company pays £13 per hour. Believe it or not, there are two quantities to consider here. You have to consider the money earned and the hours worked.

Now, we have to figure out what x and y are.

Recall that x is the input or independent quantity and y is the output or dependent quantity.

The independent quantity here is the hours worked. This is because the worker has control over it. The worker decides how many hours they can work. The dependent quantity is the money earned because it depends on how many hours are worked.

Recall the multiplicative relationship equation

$y=ax$

A worker will earn £13 per hour meaning that: $y=13$ and $x=1$. We can substitute this in the equation to get the constant,

$y=ax\phantom{\rule{0ex}{0ex}}13=1\times a\phantom{\rule{0ex}{0ex}}a=13$

For a question like this, you may not need to go through this step to find the constant because it is quite clear that you will have to multiply 13 by the number of hours worked to get the total money earned.

Now let's move on to creating a table. The table will contain the number of hours worked and the money earned. The number of hours will be multiplied by the constant 13 to get the money earned.

We have

$y=ax$

when $x=1$,

$y=13\times 1\phantom{\rule{0ex}{0ex}}y=13$

when $x=2$,

$y=13\times 2\phantom{\rule{0ex}{0ex}}y=26$

when $x=5$,

$y=13\times 5\phantom{\rule{0ex}{0ex}}y=65$

when $x=6,$

$y=13\times 6\phantom{\rule{0ex}{0ex}}y=78$

So, we will use the values we've got to fill the table.

Number of hours worked (x) | Money earned (y) |

1 | 13 |

2 | 26 |

5 | 65 |

6 | 78 |

With this information, we can now plot a graph.

We know that the graph is correct because it is linear.

## Multiplicative Relationship - Key takeaways

- Multiplicative relationships between quantities are the relationships that exist when the quantities are directly proportional to each other or they are multiples of each other.
- If you have a set of inputs and outputs and you want to determine if their relationship is multiplicative, you take note of the constant and make sure it is the same all through.
- If the constant is not the same, then it is not a multiplicative relationship.
- The constant of multiplication in a multiplicative relationship is not necessarily an integer, it can be a real number.
- The graph of a multiplicative relationship between quantities is a straight line passing through the origin.

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##### Frequently Asked Questions about Multiplicative Relationship

What is multiplicative relationship between quantities?

Multiplicative relationship between quantities is the relationship that exists when the quantities are multiples of each other.

How do you know if a relationship is additive or multiplicative?

If a relationship is additive, you will be able to add a constant number to one value to get a corresponding value all through. If a relationship is multiplicative, you will be able to multiply a constant number to one value to get a corresponding value all through.

What is an example of multiplicative relationship between quantities?

Two pairs of quantities whose values are 2 and 4 and 1 and 2 respectively have a multiplicative relationship because 2 can be multiplied by 2 to get 4 and 2 can also be multiplied by 1 to get 2.

What is a multiplicative pattern?

A multiplicative pattern is when a constant number is multiplied by an input to get a corresponding value.

What are the aspects of multiplicative relations?

The aspects of multiplicative relations are the input, output and the constant.

The multiplicative relation is y = ax

where y is the output

x is the input

a is the constant

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