Mathematics notes, unit 7

Coordinates, Ratio and Proportion

Textbook Mathematics Year 5 (DLP), Part 1 and Part 2, pages Part 2, 235–250

What this chapter covers

You can state the horizontal distance and vertical distance of a point from the origin on the Cartesian plane (7.1.1)
You can find the horizontal and vertical distance between two coordinates by subtracting (7.1.1)
You can state the ratio of two quantities of the same kind, and read it as "a to b" (7.2.1)
You can state a ratio of one part to the whole, and change measurements to the same unit first (7.2.1)
You can use proportion to find a value, with multiplying, dividing and the unitary method (7.3.1)
You can solve problems that use distance on a Cartesian plane, ratio and proportion (7.4.1)

Situations and numbers in these notes are based on Mathematics Year 5 (DLP), Part 2, pp.235-250. The diagrams are redrawn.

7.1Distance between two coordinates

A Cartesian plane has a horizontal axis (the x-axis) and a vertical axis (the y-axis). They meet at the origin O (0, 0). A point is written as a pair of numbers, its coordinate.

In the coordinate (x, y), x is the horizontal distance from the origin and y is the vertical distance from the origin. Always say the horizontal number first.

Cartesian plane with the parrot at (4, 0), the jetty at (0, 5) and the lion at (5, 3) — The ticket counter is at the origin. The lion is at (5, 3): 5 units horizontally and 3 units vertically from the origin.

The parrot is at (4, 0). Its horizontal distance from the ticket counter is 4 units and its vertical distance is 0.
The jetty is at (0, 5). Its vertical distance from the ticket counter is 5 units.
The lion is at (5, 3): 5 units horizontally and 3 units vertically from the origin.

Do not swap the numbers. (5, 3) is not the same place as (3, 5). Go along the x-axis first, then up.

Distance between two points

To find the distance from one point to another, subtract the two horizontal distances, then subtract the two vertical distances. Take the smaller from the larger.

Cartesian plane of a home. The red path goes from the kitchen at (4, 0) 2 units right and then 5 units up to room 2 at (6, 5) — From the kitchen (4, 0) to room 2 (6, 5): 2 units to the right and 5 units up.

Worked example: kitchen (4, 0) to room 2 (6, 5)

Place	Horizontal distance from origin	Vertical distance from origin
kitchen	4 units	0 unit
room 2	6 units	5 units
difference	6 − 4 = 2 units	5 − 0 = 5 units

The distance from the kitchen to room 2 is 2 units horizontally and 5 units vertically.

Going left and down: room 1 (2, 5) to the dining area (0, 1)

Horizontal: 2 − 0 = 2 units (to the left).
Vertical: 5 − 1 = 4 units (downwards).
The distance is 2 units horizontally and 4 units vertically.

The start and the end of the journey decide the answer. Mark both points first. Then work out how far across and how far up or down.

On the home plane, the living room is at (3, 3) and the toilet is at (4, 6). State the horizontal and vertical distance from the living room to the toilet.

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Horizontal: 4 − 3 = 1 unit. Vertical: 6 − 3 = 3 units.

7.2Ratio between two quantities

A ratio compares one quantity with another quantity of the same kind. We write it as a : b and say "the ratio of a to b".

Three scouts and five MRCS members shown as people — 3 scouts and 5 MRCS members.

Scouts to MRCS members: 3 : 5 (three to five).
MRCS members to everybody (scouts and MRCS together): 5 : (3 + 5) = 5 : 8.
Everybody to the scouts: 8 : 3.

The order matters. 3 : 5 and 5 : 3 are different ratios. Read the question: "the ratio of A to B" means A first.

Ratio of masses

A dodol recipe uses 11 kg of granulated sugar and 10 kg of glutinous flour.

11 bars for sugar and 10 bars for flour, each bar 1 kg

Worked example

Ratio of the mass of sugar to the mass of flour = 11 : 10 (eleven to ten).
Ratio of the mass of flour to the mass of sugar and flour together = 10 : (11 + 10) = 10 : 21.

Use the same unit

You can only compare two quantities when they are in the same unit. Change the bigger unit to the smaller unit first.

Worked example: 2 m of blue ribbon and 75 cm of red ribbon

Step 1. 2 m = 2 × 100 cm = 200 cm.
Step 2. Ratio of blue to red = 200 : 75.
Writing 2 : 75 is wrong, because 2 is in metres and 75 is in centimetres.

Ask for the ratio in the order the question gives. Draw a bar for each quantity if you are not sure. The total is all the bars joined together.

A basket has 4 mangoes, 3 papayas and 6 bananas. State the ratio of papayas to all the fruit.

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Total = 4 + 3 + 6 = 13. Ratio = 3 : 13.

7.3Proportion to find a value

Quantities are in proportion when they grow or shrink together at the same rate. If you use twice as much of one, you use twice as much of the other.

A batik cake recipe uses 200 g of butter for 600 g of Marie biscuits. How much biscuit is needed for 300 g of butter?

Double number line: 200 g butter matches 600 g biscuits, 100 g butter matches 300 g biscuits, 300 g butter matches 900 g biscuits — Divide by 2 to find 100 g, then add 200 g and 100 g.

Worked example: butter 300 g

Step 1. 200 g butter for 600 g biscuits.
Step 2. Divide by 2: 100 g butter for 300 g biscuits.
Step 3. 300 g = 200 g + 100 g, so 600 g + 300 g = 900 g of biscuits.

Price and mass

Worked example: green beans cost RM8.50 for 1 kg. Price of 2 kg 400 g?

Step 1. 2 kg: 2 × RM8.50 = RM17.00.
Step 2. 100 g: RM8.50 ÷ 10 = RM0.85. So 400 g: 4 × RM0.85 = RM3.40.
Step 3. Add: RM17.00 + RM3.40 = RM20.40.

Is RM30 enough for 4 kg of barley at RM7.30 per kg? Estimate first: 4 × RM7.30 is a little less than 4 × RM7.50 = RM30, so yes. Then check: RM7.30 + RM7.30 + RM7.30 + RM7.30 = RM29.20.

Change units before you multiply. 400 g is not 4 kg. 1 kg = 1 000 g, so 100 g is one tenth of a kilogram.

1 kg of sugar costs RM2.85. Find the price of 2 kg of sugar. How much sugar can you buy for RM8.55?

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2 × RM2.85 = RM5.70. RM8.55 ÷ RM2.85 = 3 kg.

7.4Solving problems

Use the four steps: understand the problem, plan a strategy, solve, then check. Draw a Cartesian plane, a table or a bar for each quantity.

Coordinates: Resma's duty position

Jessica is at entrance A (1, 0). Resma is 4 units horizontally and 3 units vertically from Jessica (to the right and up).
Step 1. Horizontal: 1 + 4 = 5.
Step 2. Vertical: 0 + 3 = 3.
Step 3. Resma is at (5, 3).
Check: 5 − 1 = 4 and 3 − 0 = 3. Correct.

Ratio: two packages

Package R has 1 kg and package T has 4 kg. A 2 kg honeydew is put inside R.

Package R: 1 kg and 2 kg, total 3 kg. Package T: 4 kg

R now has 1 + 2 = 3 kg. Together R and T have 3 + 4 = 7 kg.
Ratio of R to the total mass of R and T = 3 : 7. The ratio of the total to T = 7 : 4.

Unitary method: petrol

A car uses 3 ℓ of petrol for a 36 km journey. How far can it go on 40 ℓ?

3 litres for 36 km, 1 litre for 12 km, 40 litres for 480 km

Step 1. Find the value for 1 unit: 36 km ÷ 3 = 12 km for 1 ℓ.
Step 2. 40 × 12 km = 480 km.
Check: 3 × 12 km = 36 km, which matches the given.

The unitary method means: find the value of one first (divide), then find the value you want (multiply).

When a quantity changes, update it before you make the ratio. R was 1 kg, but after the honeydew went in it is 3 kg. The ratio uses 3, not 1.

A printer prints 400 posters in 80 minutes. How many minutes for 700 posters?

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400 posters in 80 min, so 100 posters take 80 ÷ 4 = 20 min. 700 posters = 7 × 20 = 140 minutes.

Summary

In the coordinate (x, y), x is the horizontal distance and y is the vertical distance from the origin.
Distance between two points: subtract the horizontal distances, then subtract the vertical distances.
A ratio a : b compares two quantities of the same kind. Order matters.
Ratio of a part to the whole: add the parts to get the total first.
Use the same unit before stating a ratio (2 m and 75 cm becomes 200 : 75).
Proportion: if one quantity doubles, the other doubles. Divide to find one unit, then multiply.
Solve problems in four steps: understand, plan, solve, check.