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Sec 1 Ratio, Rate and Speed | Unit Rates, Proportion and Average Speed

Secondary 1 Ratio, Rate and Speed connects three ideas that are often taught separately but share the same structure: comparison by multiplicative relationship. A ratio compares quantities. A rate compares unlike units. Speed is a particular rate comparing distance with time.

Students searching for Sec 1 ratio rate speed, unit rate, proportion, speed formula or Secondary 1 Mathematics often memorise different procedures for each chapter. The stronger system is identify the quantities → align units → reduce to a common unit relationship → scale → check the context.

This page is the canonical Ratio, Rate and Speed owner under the Secondary Mathematics Topic Library. The Primary bridges are Units and Parts and PSLE Speed Problems.

Quick answer: the three ideas

  • Ratio: compares quantities of the same or compatible type, such as 3:5.
  • Rate: compares quantities with different units, such as $4 per kg.
  • Speed: distance per unit time, such as 60 km/h.

Ratio as multiplicative comparison

If A:B = 3:5, then for every 3 equal units of A, there are 5 equal units of B.

The numbers 3 and 5 are not actual values unless the problem gives a scale.

Equivalent ratios

3:5 = 6:10 = 9:15 because both parts are multiplied by the same factor.

Scaling only one side changes the relationship.

Simplifying ratios

Example: 18:30.

HCF of 18 and 30 is 6.

18:30 = 3:5.

Units must be compatible before simplifying. Convert metres and centimetres first if they appear in the same ratio.

Ratios with different units

Example: 2 m : 50 cm.

Convert 2 m = 200 cm.

Ratio = 200:50 = 4:1.

Never simplify mixed units directly.

Sharing in a ratio

Problem: $240 is shared in the ratio 3:5.

Total units = 8.

One unit = 240 ÷ 8 = 30.

Shares = $90 and $150.

Rate: per-one thinking

A rate connects unlike units.

Example: 12 litres fill 3 containers equally. Rate = 12 ÷ 3 = 4 litres per container.

Unit rates make comparison easier because both situations are reduced to the same “per 1” basis.

Worked rate comparison

Shop A: 6 notebooks for $15.
Shop B: 8 notebooks for $18.

Shop A unit price = 15 ÷ 6 = $2.50 per notebook.

Shop B unit price = 18 ÷ 8 = $2.25 per notebook.

Shop B is cheaper per notebook.

Speed as a rate

  • Distance = speed × time
  • Speed = distance ÷ time
  • Time = distance ÷ speed

The relationship is unchanged from Primary Mathematics; Secondary work expects stronger unit control and more efficient representation.

Unit conversion in speed

If speed is in km/h, time must be in hours and distance in kilometres unless a conversion is made.

Example: 90 minutes = 1.5 hours.

A journey of 72 km in 1.5 h has speed 48 km/h.

Average speed

Average speed = total distance ÷ total time.

It is generally not the arithmetic mean of two speeds.

If a traveller covers equal distances at 30 km/h and 60 km/h, the slower stage lasts longer, so the average is less than 45 km/h.

Proportion: same ratio, different scale

If 4 kg of rice costs $14, then 10 kg at the same rate costs:

Unit rate = 14 ÷ 4 = $3.50/kg.

10 kg cost = 10 × 3.50 = $35.

The relationship can also be expressed as equivalent ratios.

Direct proportion intuition

If one quantity doubles and the other doubles under the same rate, the relationship is directly proportional.

At Sec 1, this intuition prepares students for later algebraic and graphical representations of proportional relationships.

Scale drawings and maps

Scale is a ratio between drawing length and actual length.

Example: Scale 1:50 means 1 cm on the drawing represents 50 cm in reality.

Keep units consistent before applying the ratio.

Common errors

  • simplifies ratios before converting units
  • adds ratio parts when the problem gives a difference instead of a total
  • compares rates without reducing to common units
  • uses arithmetic mean for average speed
  • mixes minutes and hours
  • treats ratio numbers as actual quantities
  • forgets units in rate answers

The decision checklist

  1. Is this a ratio, a rate, or a speed?
  2. Are the units compatible?
  3. Would a unit rate make comparison easier?
  4. Is the known value a total, one part or a difference?
  5. Is the question asking for a direct value or a change?

Practice

Practice 1

Question: Simplify 24:36

Answer: 2:3

Practice 2

Question: 3 m : 75 cm

Answer: 300:75 = 4:1

Practice 3

Question: $180 shared 2:3

Answer: $72 and $108

Practice 4

Question: 15 items cost $45

Answer: $3 per item

Practice 5

Question: 150 km in 2.5 h

Answer: 60 km/h

Practice 6

Question: 60 km at 30 km/h + 60 km at 60 km/h

Answer: Average speed 40 km/h

Frequently asked questions

What is the difference between ratio and rate?

Ratio compares quantities; rate compares quantities using different units.

Why convert units first?

A ratio or rate is meaningful only when the units are compatible or clearly stated.

How do I compare prices?

Reduce each offer to a unit price or another common rate.

How do I find average speed?

Total distance divided by total time.

Where does this sit in Atlas?

This is the canonical Sec 1 Ratio, Rate and Speed owner under Secondary Mathematics Topic Library.

The final Ratio–Rate rule

Reduce the relationship to compatible units and a common scale. Ratio, rate and speed become one system when students see that each describes how one quantity changes relative to another.

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