Today, we’re going to learn about **inverse proportionality between magnitudes.**

To start off, we need to remind ourselves that a magnitude is anything that can be measured.

If this doesn’t ring a bell, look over our previous post where we talk about direct proportionality and explain the concept of magnitude: Direct Proportionality

##### Lots of magnitudes are related to others, for example:

- The amount of toys that you have with the amount of space that they take up.
- The speed of a car with the time it takes to travel a distance.
- The size of your room with the time it takes you to clean it.
- The time a dish spends in a hot oven with how hot the dish becomes.

We’ve already seen in our **direct proportionality** introduction that there are relationships where however much one magnitude grows the other does as well.

**But when one magnitude grows and the other shrinks proportionally, it’s called Inverse Proportionality.**

Two magnitudes are inversely proportional when one magnitude is multiplied (or divided) by a number and the other magnitude is divided (or multiplied) by the same number.

The faster a racecar…

…the less time it’ll take to finish the circuit!

Let’s imagine that to complete a lap at 100 miles/hour, it takes a car 12 min. In this example, and knowing that an inverse proportional relationship exists, we can say that if we multiply the speed by 2 (200 miles/hour), it would take half the time to finish the lap (**6 min**).

On the other hand, if the speed was halved (100 miles/hour ÷ 2 = 50 miles/hour) the lap time would be doubled (12 min x 2 =** 24 min**)

**If it took the racecar 4 min to finish its last lap, what would happen to the speed of the car during its last lap?**

(12 min ÷ 4 min = 3)

The time is **divided** by 3, so the speed needs to be **multiplied** by 3 (3 x 100 miles/hour = 300 miles/hour). That is to say, the racecar was going at a speed of **300 miles/hour in its last lap.**

##### Inverse Proportionality

With these examples, we can see this type of proportionality is called **INVERSE**. What happens to one of the magnitudes is the **INVERSE** of the other magnitude; when one increases the other decreases and vice versa.

In order to calculate the proportional reasoning, we need to multiply the quantities of each magnitude by each other.

- 100 miles/hour x 12 min = 1200
- 200 miles/hour x 6 min = 1200
- 50 miles/hour x 24 min = 1200
- 300 miles/hour x 4 min = 1200

By looking at this, we’re reminded that the proportionate reasoning is a **constant**; it’s always the same for each pair of numbers that represent the magnitudes that are being compared. In this example, the proportional reasoning is **1200**.

Remember that at Smartick you have plenty of inverse and direct proportion exercises and problems for you to practice.

### Learn More:

- Inverse Proportionality: The Rule of Three Inverse
- Direct Proportions: What Are They? What Are They Used For?
- Proportionality Problems: Learn How to Solve Them
- Compound Rule of 3: When to Use It and Some Problems
- Difference between Mathematical Constants and Variables

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