Q: What does IP Maths Notes (Lower Sec, Year 1-2): 03) Ratio, Rate & Percentage Reasoning cover? A: Structure ratio simplifications, rate conversions, percentage change chains, and mixture questions with full working.
The core idea is simple: Ratio, rate, and percentage questions are about consistent parts, units, and multipliers.
Use it as a working check: Simplify ratios by scaling all parts equally, convert rates with units visible, and handle percentage change with multipliers such as 1.20 or 0.85.
Then go one layer deeper: Follow the examples to practise average speed, reverse percentage, and mixtures, then check each answer by asking whether the size and unit are reasonable.
Mastery of ratio and percentage reasoning saves marks across chemistry mixtures, physics rates, and finance contexts. This post trains systematic setups and calculator-free checks.
These notes align with MOE Lower Secondary Mathematics syllabus used in IP pathways (aligned to O-Level Mathematics 4052 foundations).
Status: MOE Lower Secondary Mathematics syllabus (latest release) checked 2025-11-30 - scope unchanged; remains the reference for these lower-sec notes.
Write the rate formula with units before substituting numbers.
Percentage change
Original amount
Convert each percentage change into a multiplier.
Mixture questions
Amount of ingredient, such as sugar or pure metal
Track the ingredient amount before dividing by the final total.
This map prevents the two common errors: averaging rates directly and applying a percentage change to the wrong base.
1 Ratio fundamentals
1.1 Simplification & scaling
Divide all parts by the highest common factor to simplify.
To increase a ratio, multiply each part by the same scale factor.
Example: The ratio of red:blue marbles is 18:24. Simplify: divide by 6 → 3:4.
1.2 Part-whole conversion
If the ratio of boys to girls is 5:4, the total number of parts is 9. Boys make 95 of the class.
Ratio wording checkpoint
Before converting a ratio into a fraction, decide whether the comparison is part-to-part or part-to-whole.
Wording clue
Setup move
What to write
Common trap
"boys:girls = 5:4"
Compare one group with another group.
Boys to girls is 5:4.
Saying boys are 5/4 of the class.
"fraction of the class who are boys"
Add all parts first.
Boys are 5/(5+4)=5/9 of the class.
Using 5/4 because the ratio was written as boys:girls.
"boys make up 5/9 of the class"
Convert part-whole back to parts.
Boys:girls is 5:(9−5)=5:4.
Treating 5 and 9 as the two ratio parts.
Worked check: in a class where boys:girls is 5:4, boys are not 5/4 of the class because that would be more than the whole class. The class has 9 equal parts, so the boys form 5/9 of the total.
Misconception check: the colon compares listed parts; a fraction usually compares one part with the whole named group. Always name the denominator before writing the fraction.
2 Rate conversions
Common rate templates:
Speed: speed=timedistance.
Density: density=volumemass.
Unit cost: cost per unit=quantitytotal cost.
Rate unit checkpoint
Before substituting into a rate formula, make the units match the unit required in the answer. A correct fraction can still give the wrong rate if the time, distance, mass, or quantity unit is inconsistent.
Required answer unit
Convert first
Setup check
Trap to avoid
km h−1
Minutes to hours
speed=km÷h
Dividing kilometres by minutes, then labelling the answer as km h−1.
m s−1
Kilometres to metres and hours to seconds
speed=m÷s
Converting only one of the two units.
Dollars per kg
Grams to kg
unit cost=dollars÷kg
Comparing a 500 g price directly with a 1 kg price.
Grams per cm3
Volume to cm3 if needed
density=g÷cm3
Worked check: 18 km in 40 min is not 18÷40=0.45 km h−1. Since 40 min =32 h, the speed is 18÷32=27 km h−1.
Misconception check: "per hour" means one hour is the denominator unit. Convert the time into hours before dividing, or convert the final per-minute rate by multiplying by 60.
Average-rate checkpoint
For a whole journey, average speed is a fresh rate for the combined trip, not the average of the separate speeds.
Journey clue
First setup
Calculation move
Common trap
Distances and times for each part are given
Add all distances and add all times.
average speed=total timetotal distance.
Averaging the separate speeds directly.
Same time is spent at two speeds
The two time intervals are equal.
The mean of the two speeds works only because the times are equal.
Using this shortcut when the times are different.
Same distance is travelled at two speeds
The two distances are equal, not the times.
Find each time, then use total distance over total time.
Averaging the two speeds just because the distances match.
One part is in minutes and another is in hours
Time units must match before adding.
Convert all times to the required final unit.
Adding 30 minutes and 1 hour as 31.
Worked check: a runner covers 3 km at 6 km h−1 and another 3 km at 12 km h−1. The distances are equal, but the times are not: 3/6=0.5 h and 3/12=0.25 h. The average speed is
0.75 h6 km=8 km h−1.
Misconception check: (6+12)/2=9 is wrong here because the runner did not spend equal time at both speeds.
Worked example - Multi-rate problem
A cyclist travels 18 km in 40 minutes on level ground and 12 km in 35 minutes uphill. What is the average speed for the combined journey?
Total distance: 18+12=30 km.
Total time: 40+35=75 minutes=1.25 h.
Average speed: 1.2530=24 km h−1.
Trap check: do not average 18/40 and 12/35 directly. Average speed for a whole journey is always total distance divided by total time.
3 Percentage techniques
Percentage base checkpoint
Before using a percentage, name the base amount. The same percentage can mean different absolute changes when the base changes.
Question cue
Base amount
Setup
"Increase 80 by 15\%"
Original amount, 80
80×1.15
"After a 15\% discount, the price is 204"
Original price before discount
0.85P=204
"65\% rises to 70\%"
Starting percentage, 65\%
Percentage change =6570−65×100%
"65\% rises to 70\%, in percentage points"
No scaling base needed
Difference =70−65=5 percentage points
Misconception check: the word "percent" does not always mean "divide by the final amount". In most change questions, the percentage is measured from the starting or original amount.
3.1 Successive percentage changes
To apply a 20% increase then a 15% decrease to an amount A:
A×1.20×0.85=1.02A.
Net effect is a two percent increase.
Percentage multiplier direction checkpoint
For each percentage question, decide whether you are moving forward from the original amount or backward from the final amount.
Question type
Multiplier equation
What to do
Common trap
Increase by p%
Final = original ×(1+100p)
Multiply by a number greater than 1.
Adding p directly instead of p% of the original.
Decrease by p%
Final = original ×(1−100p)
Several changes in a row
Final equals original multiplied by each multiplier in order
Multiply the multipliers; do not add the percentages.
Saying +20\% then -20\% returns to the original.
Original amount is unknown
Original equals final divided by the multiplier
Divide by the multiplier that produced the final amount.
Multiplying again and making the original even smaller after a discount.
Worked check: a jacket is increased by 10%, then discounted by 10%. The net multiplier is 1.10×0.90=0.99, so the final price is 99% of the original, not the original price.
Misconception check: opposite-looking percentage changes do not cancel unless the base amount is the same. In a chain, the second percentage acts on the new amount.
3.2 Reverse percentages
If an item costs $204 after a fifteen percent discount, the original price P satisfies 0.85P=204.
P=0.85204=240.
3.3 Percentage point vs percentage change
Percentage points describe absolute difference (e.g., 65% to 70% is +5 percentage points).
To combine two solutions of different concentrations:
Let m1,m2 be masses (or volumes).
Let c1,c2 be concentrations.
Resulting concentration: m1+m2m1c1+m2c2
Mixture equation checkpoint
For mixture questions, conserve the ingredient amount, not the percentage. The total amount changes when material is added or removed, so the final percentage must be applied to the final total.
Question clue
Quantity to track
Equation pattern
Common trap
Two mixtures are combined
Amount of pure ingredient from each mixture
initial ingredient amounts add to final ingredient amount
Averaging the percentages without using the mixture sizes.
Extra syrup, water, or metal is added
Ingredient amount before and after the addition
old ingredient plus added ingredient equals final percentage times final total
Applying the target percentage to the old total only.
Some mixture is removed first
Ingredient amount remaining after removal
remaining ingredient plus added ingredient equals final ingredient amount
Removing volume but forgetting it also removes ingredient.
Final concentration is given
Final total amount
ingredient amount equals final percentage times final total
Treating the final percentage as an amount by itself.
Worked check: if 250 ml of an 8% drink is mixed with x ml of 15% syrup to make a 10% drink, the sugar equation is 0.08(250)+0.15x=0.10(250+x). The right side uses 250+x because the final drink is larger than the original drink.
Misconception check: adding syrup changes both the amount of sugar and the total volume. You cannot fix the concentration by adding 2% of the original volume.
Worked example - Alloy blend
A jeweller mixes 120 g of 18-karat gold (75% pure) with 80 g of 14-karat gold (58.3% pure). What is the purity of the alloy?
Mass of pure gold: 120×0.75+80×0.583=90+46.64=136.64 g.
Total mass: 200 g.
Purity: 200136.64=0.6832=68.32%.
Worked example - Sugar mixture setup
A 250 ml drink is 8% sugar. How much 15% sugar syrup must be added to make the final mixture 10% sugar?
Let x be the volume of syrup added, in ml.
Track sugar, not just total volume:
Source
Volume
Sugar fraction
Sugar amount
Original drink
250
0.08
0.08×250=20
Syrup added
x
0.15
0.15x
Final drink
250+x
0.10
0.10(250+x)
Set sugar amount before equal to sugar amount after:
20+0.15x=0.10(250+x).
20+0.15x=25+0.10x⇒0.05x=5⇒x=100.
So 100 ml of syrup must be added.
Common trap: do not add 2% of 250 ml. The added syrup also increases the final volume, so the target sugar amount changes.
Practice Quiz
Review ratio simplification, rate conversions, successive percentages, and mixture reasoning with the quiz below.
Try it yourself
Siobhan and Amir share money in the ratio 7:9. If Amir has $63, how much does Siobhan have?
A car travels at 72 km h−1 for 35 minutes then at 54 km h−1 for 50 minutes. Find the average speed for the journey.
The population of a town grows by 12% one year and shrinks by 5% the next. What is the net percentage change?
A 250 ml drink is 8% sugar. How much 15% sugar syrup must be added to reach a 10% mixture?
