Consider the values in each table. State whether they could represent a directly proportional relationship between x and y:
| x | 1 | 3 | 5 | 7 |
|---|---|---|---|---|
| y | 50 | 40 | 30 | 20 |
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| y | 5 | 20 | 45 | 80 |
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| y | 5 | 10 | 15 | 20 |
| x | 1 | 5 | 6 | 20 |
|---|---|---|---|---|
| y | 100 | 75 | 50 | 25 |
For each of the following graphs, state whether they could represent a directly proportional relationship between x and y:
Describe how the two quantities that are in direct proportion change.
Determine whether the following are examples of direct variation between x and y:
When x is doubled, y is halved.
The y value is always 4 times the x value.
When x increases by 50\%, y increases by 50\%.
When x increases by 6, y increases by 6.
Suppose a is directly proportional to b. When b = 6, a = 60.
Write an equation for a in terms of b.
Graph the relationship between a and b for 0 \leq b \leq 10.
State the gradient of the line on the graph.
Find the value of a when b = 3.
Find the value of b when a = 70.
State whether the following equations represent a direct variation between the variables:
C = 300 p
C = 80 n
y = - 6 x
y = 2 + 0.2 x
C = 90 n + 1
y = 2 x + 8
Two quantities are directly proportional. If one quantity increases by a multiple of 7, what happens to the other quantity?
Suppose x varies directly with y, and x = 12 when y = 8. Find the value of x when y = 16.
Consider the proportional relationship shown in the table:
Graph the proportional relationship represented in the table.
Find the value of y when x = 8.
| x | 10 | 20 | 30 | 40 | 50 |
|---|---|---|---|---|---|
| y | 9.5 | 19 | 28.5 | 38 | 47.5 |
Suppose the constant of variation, k, is positive and y varies directly with x.
Describe how y changes as x increases.
Describe how y changes as x decreases.
For each of the following, write an equation, using k as the constant of variation:
m varies directly with p.
The revenue of a company, r, varies directly with the number of sales, n.
Find the equation relating the following variables given the table of values:
| a | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| b | 0 | 2 | 4 | 6 |
| n | 0 | 2 | 4 | 6 |
|---|---|---|---|---|
| r | 0 | \dfrac{8}{5} | \dfrac{16}{5} | \dfrac{24}{5} |
Consider the equation P = 90 t.
State the constant of proportionality.
Find the value of P when t = 2.
Suppose y varies directly with x, and y = \dfrac{1}{5} when x = 4.
Find the constant of variation, k.
Hence, find the equation of variation of y in terms of x.
A car travelling at a constant speed travels d \text{ km} in t hours.
Write an equation relating d and t, where k is the constant of variation.
If the car travels 360 \text{ km} in 4 hours, find the value of k.
Hence, write the equation for d in terms of t.
How far will the car travel in 12 hours?
The number of revolutions a wheel makes varies directly with the time it rolls for. A bike wheel revolves r times in t seconds.
Write an equation relating r and t, where k is the constant of variation.
If the wheel completes 40 revolutions in 8 seconds, find the value of k.
Hence, write the equation for r in terms of t.
How many revolutions will the wheel complete in 5 seconds?
How many revolutions will the wheel complete in 7 seconds?
Noah's pay, p, is directly proportional to the number of hours, h, he works. For an 8 hour day, he receives \$192.
Calculate the constant of variation.
Write a linear equation for p in terms of h.
Calculate Noah's pay if he works for 18 hours.
How many hours does Noah need to work to earn \$336?
The height, h, of a regular tetrahedron varies directly with its side length, s. A particular tetrahedron with a side length of 5 \text{ cm} has a height of 7.07 \text{ cm}.
Find the constant of variation, k, to two decimal places.
Write an equation for h in terms of s.
Find the height of a different tetrahedron with a side length of 2 \text{ cm}.
The original of a printed image measures 8.5\text{ cm} in width and 34\text{ cm} in length. When a customer wants to print a copy in a different size, the width and length must be in the same ratio as the original so that the photo does not appear distorted.
If x represents the width and y represents the length of the printing size, write the equation for y in terms of x.
Find the length of a copy if the width of the copy is 13 \text{ cm}.
Luke works 3 days a week and earns \$900 per day.
How much does he earn in 5 weeks?
How much does Luke earn if he works for 7 days in one week?
Write an equation for the amount earned, y, if Luke works for x days.
Pauline earns \$178.97 in 11 hours. How much will she earn in 19 hours?
A quad bike travels 3.4 \text{ km} in 17 minutes at a constant speed. How long would it take to travel 1.8 \text{ km}?
3.1 \text{ kg} of pears cost \$7.13. How much would 1.5 \text{ kg} cost?
A boat can travel 35.75 \text{ km} on 16.25 \text{ L} of petrol. How much petrol will it need to travel 34.32 \text{ km}?
4.2 \text{ kg} of pears cost \$7.14. How many kilograms of pears can you purchase with \$6.46?
A quad bike travels 2.8 \text{ km} in 14 minutes at a constant speed. How far would it travel in 1.3 hours at this same speed?
Uther earns \$423.13 in 17 hours. How many hours does he work in a week if he earns \\ \$273.79?
If 172 \text{ mL} of a medicine must be mixed with 400 \text{ mL} of water, how many millilitres of this medicine must be mixed with 1593 ounces of water? Round your answer to the nearest millilitre.
The weight of an object on the moon varies directly with the weight of the object on the Earth. An astronaut who weighs 660 \text{ N} on Earth weighs only 110 \text{ kg} on the moon.
A lunar landing craft weighs 18\,500 \text{ N} when on the moon. Calculate the weight of this landing craft when on Earth.
The amount of petrol used by a car travelling at a constant speed varies directly with the distance travelled.
A car uses 12 \text{ L} of petrol to travel 100 \text{ km}. How much petrol will be used to travel 150 \text{ km} at the same speed?
The amount of white and red paint needed to make 'Flamingo Pink' coloured paint is shown in the following graph, where x is the amount of white paint and y is the amount of red paint needed.
Write an equation relating x and y.
For every 1 can of white paint, how many cans of red paint is required to make flamingo pink?
The colour 'Sunset Red' requires the amount of red paint to be 20 times the amount of white paint.
Would the graph for 'Sunset Red' be flatter or steeper than the graph for 'Flamingo Pink'?
An empty box weighs 1 \text{ kg}.
How much will it weigh if it contains a load of:
Write an equation for the total weight of the box, w, if it has an l \text{ kg} load.
Graph the relationship between l and w for 0 \leq l \leq 10.
The table shows the cost of various amounts of petrol per litre:
| \text{Number of litres }(x) | 0 | 10 | 20 | 30 | 40 | 50 |
|---|---|---|---|---|---|---|
| \text{Cost of petrol }(y) | 0 | \$16 | \$32 | \$48 | \$64 | \$80 |
How much does petrol cost per litre?
Write an equation relating the number of litres of petrol, x, and the cost of the petrol, y.
In this equation, what does the coefficient of x represent?
How much would 65\text{ L} of petrol cost at this price?
Graph the relationship between x and y.