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.
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.
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.