Junior Maths | Percentages and Ratio Prep Guide (for tutors) | Change, points and sharing in three steps

2026-11-05

Tutor lesson-prep guides · Secondary Maths

Ask what happens when a price rises 15 per cent and then falls 20 per cent and most students say nothing changes. This lesson prices one item three times so they discover that percentages with different bases cannot be added.

Objectives — Students can: (1) tell percentage points from per cent, explaining that 20 to 25 is five points and a 25 per cent relative rise; (2) compute successive changes and explain why they do not add; (3) share a total in the ratio 3 to 5 and verify it.

01 Prerequisites (3-minute check)

If any of these fails, fix it before the lesson starts.

Skill	Quick check	Pass mark
Fractions and decimals	0.15 as a fraction	15 over 100, that is 15 per cent
Basic percentage	25 per cent of 200	50
Ratio idea	How many parts in 3 to 5	8 parts

02 Lesson flow (60 minutes)

Time	Stage	Tutor does	Student does
0-8 min	Contradiction opener	Price 480, up 15 per cent, down 20 per cent: back to 480?	Guess and justify
8-20 min	Points versus per cent	20 to 25 is 5 points and a 25 per cent rise	Say it both ways
20-34 min	Successive change	480 to 552 to 441.6, net -8 per cent	Multiply twice
34-46 min	Discount and reverse	300 less ten per cent is 270; reverse it	Work backwards
46-56 min	Ratio sharing	240 in 3 to 5 gives 90 and 150	Check the sum
56-60 min	Exit check	Three questions collected and marked	Work alone

03 Teaching script (read it out)

Scene 1: Two languages: points and per cent

Going from 20 to 25 is a rise of 5 percentage points, but a relative rise of 25 per cent, because 5 divided by 20 is a quarter.
Newspapers, report cards and polls mix the two constantly. Have students say the same fact both ways; a wrong version is audible immediately.

Students say: a pass rate rising from 60 to 72 per cent, once in points and once in per cent.

Scene 2: Successive changes never add

A price of 480 rises 15 per cent to 552; the next 20 per cent cut is taken from 552, giving 441.6, a net -8 per cent.
The base moved, so the second cut is larger in dollars than a cut from 480 would be. Note the mirror case: cost 320, price 400, profit 80, margin 25 per cent on cost only.
Every percentage needs its base named before the arithmetic starts.

Students do: write three lines and label the base on each line.

Scene 3: Ratio sharing: parts first, then multiply

3 to 5 is 8 parts, so 240 divided by 8 is 30 per part, giving 90 and 150.
Always verify twice: the two shares must add to the total, and their quotient must equal the ratio. Students who check only the sum never catch a reversed ratio.

Students do: repeat with 360 and verify both ways.

04 Question types and worked steps

Type	Example	Steps	Answer
Points versus per cent	Describe 20 to 25	Name the base first	5 points, 25 per cent relative
Successive change	480 up 15 then down 20	Multiply each time	441.6, net -8 per cent
Reverse a discount	Ten per cent off leaves 270	Divide, do not add back	300
Ratio sharing	240 in 3 to 5	Parts, divide, multiply	90 and 150

Sanity check (teach this): Sanity checks: up 15 and down 20 does not return to the start because the second base is 552 and not 480; the net change is -8 per cent, not plus 5 or zero; cost 320 with price 400 is a 25 per cent margin, not 20; reversing 270 needs division by 0.9 to reach 300, while adding ten per cent gives only 297.

05 Independent practice (3 questions)

Question	Answer	What to watch
Two languages	60 to 72 per cent	Points and per cent kept apart
Two successive changes	250 up 28 then down 10 per cent	Second base differs
Reverse a price	20 per cent off leaves 400	Divide by 0.8, not times 1.2
Ratio with check	360 in 3 to 5	Shares add to the total

06 Common mistakes and what to say

Mistake	Student writes	Say this
Adding percentages	Up 15 then down 20 means down 5	The second 20 per cent is taken from 552, not 480. Different base, so the rates do not add.
Adding back a discount	Ten per cent off 270 means 297	A discount multiplies by 0.9, so reversing divides by 0.9; adding assumes an unchanged base.
Points read as per cent	The pass rate rose 12 per cent	From 60 to 72 is 12 points, which is a 20 per cent relative rise.
Ratio reversed	3 to 5 split as 150 and 90	Which side is larger? State what 3 and 5 stand for first.

07 Homework (paper, section and question numbers)

Secondary Maths chapter index — Every chapter in the subject
All prep guides — Parallel chapters
Redo this chapter — Two successive-change items and one ratio item, all verified

08 Exit check (last 5 minutes)

Question	Answer	Decision
20 to 25 per cent	5 points, a 25 per cent rise	Rising 5 per cent means restart act one
480 up 15 then down 20	441.6, net minus 8 per cent	480 means redo act two
240 in 3 to 5	90 and 150	Reversed means redo act three

09 Tutor FAQ

Must students work without a calculator?

Through S2 the decimal work should be by hand, with the calculator used to check. Successive changes need two written lines.

Is the phrase percentage point examinable?

Data questions in other subjects use it constantly, and misreading it wrecks the whole item, so five minutes here pays.

May ratios be simplified?

Yes, but then say what one part is worth; multiply-then-divide produces fewer errors.

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A TutorZone lesson-prep guide. Written by the TutorZone team; questions and lesson flow are original and are not taken from any school or publisher material.