Direct proportion
🎯 In this topic you will
- Explain what it means for quantities to be in proportion.
- Understand that when one quantity increases or decreases, other related quantities change in the same ratio.
🧠 Key Words
- direct proportion
- enlarge
- proportion
Show Definitions
- direct proportion: A relationship between two quantities where they increase or decrease together in the same ratio.
- enlarge: To increase the size of an object or figure while keeping its shape the same.
- proportion: A mathematical relationship showing that two ratios or quantities are equal.
🖼️ Changing the Size of Photos
Have you ever taken a photograph and liked it so much that you wanted a bigger copy to put on your wall or a smaller copy to stick in a notebook? In mathematics, we call these larger and smaller versions enlargements.
📏 Shapes That Stay in Proportion
The lengths of lines in the photo and the enlargement are in proportion, and all the angles stay the same size. In this section, you will learn about shapes and objects that are in proportion.
❓ EXERCISES
1. $3$ melons cost $\$2$. What is the cost of $15$ melons?
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2. Magda organises a meal for $12$ people. She buys $1$ pizza for every $3$ people. How many pizzas does Magda buy?
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3. Here is a recipe for ice cream.
Kiki makes ice cream for $4$ people. Write a list of the ingredients she uses.
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4. A teacher buys $24$ posters for his classroom. He can buy $4$ posters for $\$7$. How much does the teacher spend on posters?
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5. The length of a model car is one-tenth the size of the real car.
a. The model car is $40$ cm long. What is the length of the real car?
b. The real car is $150$ cm high. How tall is the model car?
c. A wheel on the model car has diameter $4.25$ cm. What is the diameter of a wheel on the real car?
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🧠 Reasoning Tip
Remember that a scale of $1:24$ means the real car is $24$ times longer than the model car.
6. You will need a calculator for this question. Dimitri has five model cars. He knows they are built to a scale of $1:18$ or $1:24$ or $1:32$ but he does not know which scale has been used for each car. He measures the length of the model cars: Beetle $170$ mm, Puma $230$ mm, Delta $140$ mm, Embla $190$ mm, Modi $160$ mm. This table shows the lengths of the real cars.
| Car | Beetle | Puma | Delta | Embla | Modi |
|---|---|---|---|---|---|
| Length in mm | $4080$ | $4140$ | $4480$ | $4560$ | $5120$ |
Work out the scale used for each car and copy and complete the sorting diagram.
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7. Here is rectangle A.
Rectangle A is enlarged to make rectangles B, C and D. Copy and complete the table.
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8. Draw a rectangle $2$ cm by $4$ cm. Label the rectangle A.
a. Draw a rectangle B so the ratio of the lengths $A : B$ is $1 : 2$.
b. Draw a rectangle C so the ratio of the lengths $A : C$ is $2 : 1$.
c. Find the perimeter of rectangles A, B and C.
d. What do you notice about the ratio of the perimeters $A : C$?
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🧠 Think like a Mathematician
A series paper sizes
Look at the table of measurements of paper sizes.
| Size | Width × height (mm) |
|---|---|
| A0 | $841 \times 1189$ |
| A1 | $594 \times 841$ |
| A2 | $420 \times 594$ |
| A3 | $297 \times 420$ |
| A4 | $210 \times 297$ |
| A5 | $148 \times 210$ |
Follow-up Questions:
👀 show answer
- 1: Each paper size is obtained by halving the previous size while keeping the same proportions.
- 2: The ratio $\text{height} \div \text{width}$ is almost the same for every size (about $1.414$). This shows that all A-series paper sizes have the same shape.
- 3: A6 is half the size of A5. Width becomes $105$ mm and height becomes $148$ mm. So the height of A6 paper is $148$ mm.