P R E P A R E D B Y :

C H U A , E R N E S T O E R I C I I I A .C O N C I O , M I C H A E L A

G O N Z A L E S , M A . I R E N E G .

2 E 2 - 2 M M

Variations

EXPLORE: RELATED QUANTITIES

Which of the following quantities are related?

DIRECT VARIATION

Let us consider the relationship between a car’s speed and the distance it covers in a given time.

Example 1

A car travels at 60 kilometers per hour a) After 2 hours, how far has it traveled?

b) After 3 hours, how far has it traveled?

c) How long will it take the car to travel a distance of 240 km? 300 km?

DIRECT VARIATION

Let us organize the answers to the questions in this table.

Notice that as the time increases, the distance increases. Asthe time decreases, the distance decreases. Thus, in thisexample, time and distance are directly related.

The table shows an ordered pair of numbers, all having thesame ratio of distance over time, which is 60.

(2, 120), (3, 180), (4, 240), (5, 300) and so on.

Time (hr) 2 3 4 5

Distance (km) 120 180 240 300

DIRECT VARIATION

Let x and y denote two quantities. y varies directly with x, or y is directly proportional to x, if there is a nonzero number k such that:

y = kx

The number k is called the constant of proportionality or the constant of variation.

Explore: SPEED AND TIME

Jeffrey and Joy are siblings who live in the same house and go to school by the same route.

1. Joy walks to school while Jeffrey runs. Who will each the school first? Why?

2. If Joy and Jeffrey ran to school at the same times, would they reach the school at the same time? Justify your answer.

3. How does the speed at which one travels relate to the time it takes to travel a certain distance?

INVERSE VARIATION

Example 2A car is traveling a distance of 120 kilometers. How longwill it take the car to reach its destination if it travels at a speed of 20 kph? 40 kph? 60 kph? 80 kph?

Solution: Since t=d/r, we can have the following values.

Notice that as the rate increases, the time decreases. As the time increases, the rate decreases. Thus, in this example, rate and time are inversely related.

rate (kph) 20 40 60 80

time (hr) 6 3 2 1 1/2

INVERSE VARIATION

Let x and y denote two quantities. y varies inversely with x, or y is inversely proportional to x, if there is a nonzero constant k such that:

y = k/x

The number k is called the constant of variation.