Mathematics notes, unit 7
Coordinates, Ratio and Proportion
Textbook Mathematics Year 6 SK (DLP, Semakan 2017), pages 187–206
What this chapter covers
- You can find the horizontal distance and vertical distance between two points on the Cartesian plane (7.1.1)
- {'You can use a scale, such as 1 cm represents 1 km or 1': '100 000, to find actual distances (7.1.1)'}
- You can state the ratio of two quantities and write it in the simplest form (7.2.1)
- You can convert units before finding a ratio (7.2.1)
- You can find the missing quantity when a ratio is given, using a bar model (find the value of one part first) (7.3.1)
- You can solve problems that combine coordinates, ratio and proportion (7.4.1)
7.1Distance between two coordinates
A point on the Cartesian plane is written as (x, y). The first number is how far across (horizontal) and the second is how far up (vertical) from the origin O. A scale tells us what one unit on the plane stands for in real life.
Horizontal distance: the difference between the two x values.
Vertical distance: the difference between the two y values.
Always take the smaller number from the larger.
Vertical distance: the difference between the two y values.
Always take the smaller number from the larger.
Worked example: distance between two places
Aina's house is at (1, 4). The market is at (4, 2).Step 1. Horizontal distance = 4 − 1 = 3 units = 3 km.
Step 2. Vertical distance = 4 − 2 = 2 units = 2 km.
So the market is 3 km horizontally and 2 km vertically from Aina's house.
Step 3. Library (5, 4) from Aina's house: horizontal 5 − 1 = 4 km, vertical 4 − 4 = 0 km. They are on the same horizontal line.
Using a scale
The scale may not be 1 unit = 1 km. Always read the scale first. Multiply the units by what one unit stands for.
Worked example: a scale of 1 unit = 2 km
Step 1. Horizontal: 4 − 1 = 3 units. 3 × 2 km = 6 km.Step 2. Vertical: 3 − 1 = 2 units. 2 × 2 km = 4 km.
The lake is 6 km horizontally and 4 km vertically from the farm.
A scale of 1 : 100 000 means 1 cm on the plane is 100 000 cm in real life.
100 000 cm = 1 000 m = 1 km. So 1 cm represents 1 km.
100 000 cm = 1 000 m = 1 km. So 1 cm represents 1 km.
Do not forget the scale. "3 units" is not the real distance. Also, do not add the x values: subtract to find the distance between two points.
On a plane with scale 1 cm : 1 km, point H is at (2, 5) and point J is at (6, 2). How far is J from H horizontally and vertically?
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Horizontal: 6 − 2 = 4 km. Vertical: 5 − 2 = 3 km.
7.2Ratio between two quantities
A ratio compares the size or value of one quantity with another. The ratio of 6 mangoes to 4 mangosteens is written 6 : 4. The order matters: "mangoes to mangosteens" is not the same as "mangosteens to mangoes".
To write a ratio in its simplest form, divide both numbers by the same number until no number (other than 1) divides both. This is just like simplifying a fraction. The value of the ratio does not change.
Worked example: simplest form
Simplify 12 : 18.Step 1. Divide both by 2: 12 : 18 = 6 : 9.
Step 2. Divide both by 3: 6 : 9 = 2 : 3.
Shortcut: divide by 6 straight away, the biggest number that goes into both.
Same units first
Both quantities must have the same unit before you write the ratio.
| Type | Convert | Ratio |
|---|---|---|
| Mass | 1 kg = 1 000 g | 3 kg : 500 g = 3 000 : 500 = 6 : 1 |
| Length | 1 m = 100 cm | 2 m : 50 cm = 200 : 50 = 4 : 1 |
| Money | RM1 = 100 sen | 20 sen : RM1 = 20 : 100 = 1 : 5 |
| Volume | 1 ℓ = 1 000 mℓ | 600 mℓ : 2 ℓ = 600 : 2 000 = 3 : 10 |
Worked example: part to total
Ravi buys 2 kg of chicken and 4 000 g of mutton. Find the ratio of chicken to the total mass.Step 1. 4 000 g = 4 kg. Total = 2 kg + 4 kg = 6 kg.
Step 2. Chicken : total = 2 : 6 = 1 : 3.
Do not write the units in the ratio. "2 kg : 4 000 g" is wrong; change to 2 000 : 4 000 or 2 : 4 first. Also check what the question asks: "part to total" is not "part to part".
Write 24 : 36 in the simplest form. Then write 50 sen to RM2 in the simplest form.
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24 : 36 = 2 : 3 (divide by 12). RM2 = 200 sen, so 50 : 200 = 1 : 4.
7.3Proportionate quantity
If you know the ratio and one of the quantities, you can find the other. Two ways: a bar model (find the value of one part) or multiply both sides of the ratio by the same number.
Worked example: one quantity is given
The ratio of Zikri's comics to Wong's comics is 3 : 2. Zikri read 12 comics. How many did Wong read?Method 1 (parts). 3 parts = 12 comics, so 1 part = 12 ÷ 3 = 4. Wong has 2 parts = 2 × 4 = 8 comics.
Method 2 (multiply). 3 : 2 → ×4 → 12 : 8. Wong read 8 comics.
Worked example: the total is given
Two friends cycle 36 km altogether. The ratio of the distance from home to P to the distance from P to Q is 4 : 5.
Step 1. Total parts = 4 + 5 = 9.
Step 2. 1 part = 36 km ÷ 9 = 4 km.
Step 3. Home to P = 4 × 4 = 16 km. P to Q = 5 × 4 = 20 km.
Check: 16 + 20 = 36 km. Yes.
Step 2. 1 part = 36 km ÷ 9 = 4 km.
Step 3. Home to P = 4 × 4 = 16 km. P to Q = 5 × 4 = 20 km.
Check: 16 + 20 = 36 km. Yes.
Extension. When the difference is given: the number of parts of the difference equals the difference in the ratio. For 7 : 4 with a difference of 90, 7 − 4 = 3 parts = 90, so 1 part = 30.
Draw the bar model first. Label what the total or difference stands for. Then find 1 part.
When the total is given, do not divide by only one of the numbers in the ratio. Divide by the total number of parts (4 + 5 = 9).
The ratio of Sara's stickers to Lim's stickers is 2 : 5. Sara has 18 stickers. How many does Lim have?
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2 parts = 18, so 1 part = 9. Lim has 5 × 9 = 45 stickers.
7.4Solving problems
Follow four steps: understand the problem, plan a strategy (drawing, table or bar model), solve, and check your answer.
Worked example: a change, then a ratio
A class has 15 boys and 18 girls. 1 boy and 2 girls move to other schools. Find the ratio of boys to girls now.Step 1. Boys now = 15 − 1 = 14. Girls now = 18 − 2 = 16.
Step 2. 14 : 16 = 7 : 8 (divide by 2).
Check: 7 × 2 = 14 and 8 × 2 = 16. Yes.
Worked example: coordinates and distance
On a plane with scale 1 cm : 1 km, Miss Norehan is at (2, 5) and Kaswini is at (5, 4).Horizontal: 5 − 2 = 3 km. Vertical: 5 − 4 = 1 km.
Kaswini is 3 km horizontally and 1 km vertically from Miss Norehan.
Ratio with a change: work out the new amounts first, then write the ratio and simplify.
There are 30 red and 20 blue beads. 10 red beads are taken out. Write the ratio of red to blue beads now in the simplest form.
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Red now = 30 − 10 = 20. Ratio 20 : 20 = 1 : 1.