Exam-Style Questions on Proportion

At depths below 900 metres, the temperature of the water in the sea is given by the formula:

Where \(x\) is the depth in metres and the temperature of the water is T °C.

(a) A fish swims downwards from a depth of 1000 metres to a depth of 1500 metres. How many degrees colder is the sea water at 1500m than 1000m?

(b) Roughly sketch the shape of a graph that shows that T is inversely proportional to \(x\).

Worked Solution

The Billing triplets are planting seedlings on the first day of the month. The three of them take two hours to plant 300 seedlings.

(a) On the second day of the month the triplets are joined by their friend Billy who helps them. Working at the same rate, how many plants should the four of them be able to plant in two hours?

(b) Working at the same rate, how much longer would it take four people to plant 1000 seedlings than it would take five people?

(c) Billy says that it took two hours for three people to plant 300 seedlings. If I assume they work all day, then in one day three people will plant 3600 seedlings because 300 × 12 = 3600.

Why is Billy's assumption not reasonable? What effect has Billy's assumption had on his answer?

Worked Solution

A Primary school asked 9 children to remove the weeds from the school gardens. The gardens measured 100m² in total and the job was completed in 10 hours (spread over many weeks).

(a) The school was asked to find volunteers to weed another garden measuring 250m² at a nursing home. This work has to be completed in 15 hours.

Calculate the least number of children the school needs to find for this work.

(b) State one assumption you have made in your answer to part (a). How would your answer to part (a) change if you did not make this assumption?

Worked Solution

People are paid to paint polygons.

There are eight people in the polygon painting posse.

All of the people paint at the same rate.

When all of the people are painting, they can paint all of the polygons in the palace in ten days.

The table shows the number of people painting each day:

Work out the total number of days taken to paint all of the polygons in the palace.

Worked Solution

The following table shows corresponding values for two variables \(x\) and \(y\).

(a) If \(y\) is inversely proportional to the square of \(x\) find an equation for \(y\) in terms of \(x\).

(b) Find the positive value for \(x\) when \(y = 20\).

Worked Solution

9 cup cakes weigh a total of 204 g.

There are 420 calories in 100 g of cup cakes.

How many calories are there in one cup cake?

Worked Solution

(a) Sketch a graph on the axes below left that shows that \(y\) is directly proportional to \(x\).

(b) Sketch the graph of \(y = -x^3\) on the axes above right.

(c) It is possible to draw many rectangles that have area \(48 cm^2\).

Draw a graph to show the relationship between length and width for rectangles with area \(48 cm^2\) and sides less than \(50cm\).

Worked Solution

At a constant temperature, the volume of a gas \(V\) is inversely proportional to its pressure \(p\). By what percentage will the pressure of a gas change if its volume increases by 15% ?

Worked Solution

Which of the following statements are correct if \(xy = c\) and \(c\) is a constant.

• (a) \(y\) is directly proportional to \(x\)
• (b) \(y\) is directly proportional to \(\frac{1}{x}\)
• (c) \(y\) is inversely proportional to \(\frac{1}{x}\)
• (d) \(x\) is directly proportional to \(y\)

Worked Solution

Henrietta sells the eggs that the chickens on her farm lay.

She has 220 chickens and each chicken lays 5 eggs each week.

Henrietta gives each chicken 80 grams of chicken feed each day which costs £5.50 for a 20 kg bag.

Work out the cost of the chicken feed for every box of six eggs she sells.

Worked Solution

If \(a\) is inversely proportional to \(b\) and \(b\) is directly proportional to \(c^2\) find a formula for \(a\) in terms of \(c\) given that \( a=20 \) and \(c = 4 \) when \(b = 8 \).

Worked Solution