Learning Objectives
- Simplify ratios and divide quantities in a given ratio.
- Use direct proportion and scale reasoning.
- Calculate with rates such as speed, pay, density and exchange rates.
- Solve average speed problems with consistent units.
Key Terms
- Ratio
- A comparison of quantities by division.
- Proportion
- A relationship in which two ratios are equal.
- Rate
- A comparison of quantities with different units.
- Unit rate
- A rate expressed per one unit.
- Average speed
- Total distance divided by total time.
- Scale
- A ratio connecting a representation to the actual object.
Simplifying Ratios
Quantities in a ratio must be in the same units before simplification. Divide every part by their HCF. Ratios have no units after comparable quantities are converted to a common unit.
Worked Example: Unit Conversion In Ratios
Question: Simplify 1.8 m : 75 cm.
- Convert 1.8 m to 180 cm.
- The ratio is 180:75.
- Divide both parts by 15.
Answer: 12:5.
Dividing A Quantity In A Ratio
Add the ratio parts to find the total number of shares. Find the value of one share, then multiply by each part.
Worked Example: Sharing
Question: Divide $1260 in the ratio 2:3:4.
- Total parts =2+3+4=9.
- One part =1260\div9=140.
- Shares are 2\times140=280, 3\times140=420 and 4\times140=560.
Answer: $280, $420 and $560.
Direct Proportion
In direct proportion, multiplying one quantity by a factor multiplies the other by the same factor. The ratio \frac{y}{x} is constant, giving y=kx. The unitary method is often simplest in context.
Worked Example: Recipe
Question: A recipe for 6 people uses 450 g of flour. How much flour is needed for 14 people?
- For one person: 450\div6=75 g.
- For 14 people: 75\times14=1050 g.
Answer: 1050 g.
Scale Drawings And Maps
A scale of 1:50 000 means 1 unit on the map represents 50 000 of the same units in reality. Convert the final answer into appropriate units.
Worked Example: Map Scale
Question: On a map with scale 1:25 000, two places are 7.6 cm apart. Find the actual distance in kilometres.
- Actual distance =7.6\times25000=190000 cm.
- Convert to metres: 190000\div100=1900 m.
- Convert to kilometres: 1900\div1000=1.9 km.
Answer: 1.9 km.
Rates
A rate compares different units, such as dollars per hour, litres per minute, people per square kilometre or kilometres per litre. Units are part of the answer and help determine whether to multiply or divide.
| Rate | Typical calculation |
|---|---|
| Hourly pay | earnings ÷ hours or rate × hours |
| Flow rate | volume ÷ time |
| Density | mass ÷ volume |
| Population density | population ÷ area |
| Fuel consumption | distance ÷ fuel, or fuel ÷ distance depending on stated unit |
| Exchange rate | multiply or divide according to the direction of conversion |
Speed, Distance And Time
The units must be consistent. Convert minutes to hours when speed is required in km/h. Average speed is based on total distance and total time; it is not generally the average of two speeds.
Worked Example: Average Speed
Question: A cyclist travels 45 km in 3 hours 45 minutes. Find the average speed.
- Convert time: 3\text{ h }45\text{ min}=3.75 h.
- Use \text{speed}=45\div3.75.
Answer: 12 km/h.
Worked Example: Two-Part Journey
Question: A car travels 120 km at 80 km/h and then 90 km at 60 km/h. Find the average speed for the whole journey.
- First time =120\div80=1.5 h.
- Second time =90\div60=1.5 h.
- Total distance 210 km; total time 3 h.
- Average speed =210\div3.
Answer: 70 km/h.
Exchange Rates
Read the rate carefully. If 1 USD = 278 PKR, convert dollars to rupees by multiplying by 278 and rupees to dollars by dividing by 278. In real transactions, a commission or separate buying/selling rate may also be included if stated.
8th Edition Chapter Map
- Ratios, including three-part and fractional ratios
- Rates and unit rates, including currency exchange
- Time, timetables and time zones
- Speed and average speed
Ratio As A Multiplicative Comparison
The ratio a:b means that the first quantity is \frac{a}{b} times the second; it does not specify their actual sizes. Quantities must use the same units before a ratio is simplified. Fractional ratios can be cleared by multiplying every term by a common denominator.
Worked Example: Fractional Ratio
Simplify \frac34:1\frac12:\frac58. Write \frac34:\frac32:\frac58 and multiply all terms by 8, giving 6:12:5.
Dividing In A Ratio
Add the ratio parts, find the value of one part and multiply. In a three-part ratio, maintain the stated order and check the final parts add to the original total.
Worked Example: Three-Way Division
Divide 9 360 in the ratio 5:3:4. Total parts =12, one part =9360/12=780. The shares are 3 900, 2 340 and 3 120.
Rates And Unit Rates
A rate compares quantities with different units, such as cost per kilogram, litres per minute or people per square kilometre. A unit rate has a denominator of one and is useful for comparing offers. Preserve the units throughout the calculation.
Worked Example: Best Value
A 1.8 kg pack costs 1 260 and a 2.5 kg pack costs 1 675. Unit costs are 1260/1.8=700 per kg and 1675/2.5=670 per kg, so the 2.5 kg pack is better value.
Currency Exchange
Read the direction of the rate carefully. If 1 USD = 280 PKR, convert dollars to rupees by multiplying by 280, and rupees to dollars by dividing by 280. In real contexts a bank may have different buying and selling rates; use the rate named in the question.
Time And Time Zones
Convert all times to a consistent format. When crossing midnight, split the interval or use a 24-hour timeline. For time zones, first identify which place is ahead. Add the offset when moving to a location ahead in time and subtract when moving to one behind, then adjust the date if necessary.
Worked Example: Journey Across Midnight
A train leaves at 22:47 and arrives at 05:18 the next day. From 22:47 to midnight is 1 h 13 min; from midnight to 05:18 is 5 h 18 min. Total duration is 6 h 31 min.
Speed And Average Speed
\text{speed}=\frac{\text{distance}}{\text{time}}. Average speed is total distance divided by total time; it is not usually the average of two speeds. Convert minutes to hours when using km/h.
Worked Example: Average Speed
A car travels 120 km at 60 km/h and 150 km at 75 km/h. Times are 2 h and 2 h. Total distance 270 km, total time 4 h, so average speed =67.5 km/h.
Extended Practice
A. The ratio of boys to girls is 7:9. There are 96 students. Find the number of girls.
Total parts 16; one part 6; girls =9\times6=54.
B. Convert 84 km/h to m/s.
84\times\frac{1000}{3600}=23.3\overline3 m/s.
C. A pump delivers 18 litres in 45 seconds. Find the rate in litres per minute.
18/45=0.4 L/s, so 0.4\times60=24 L/min.
Examination Guidance
- Convert quantities to common units before forming or simplifying a ratio.
- Include units for rates.
- For average speed, use total distance divided by total time.
- Use a reasonableness check for currency conversion: converting to a currency with more units per dollar should produce a larger numerical value.
Common Mistakes
- Simplifying 2 m : 50 cm as 2:50 without converting units.
- Dividing a total in a ratio by the largest ratio part instead of the sum of all parts.
- Averaging speeds directly when journey times or distances differ.
- Using the exchange rate in the wrong direction.
Knowledge Check
1. Simplify 2.4 kg : 600 g.
Answer
4:1.
2. Divide 945 in the ratio 3:4:2.
Answer
315, 420 and 210.
3. A 1:40 scale model is 18 cm long. Find the actual length in metres.
Answer
7.2 m.
4. A worker earns $96 for 7.5 hours. Find the hourly rate.
Answer
$12.80 per hour.
5. A train travels 156 km in 2 h 24 min. Find its average speed.
Answer
65 km/h.
Mixed Review With Full Solutions
1. Simplify 1.2\text{ kg}:450\text{ g}.
1200:450=8:3.
2. A map scale is 1:50 000. Two places are 7.6 cm apart on the map. Find the actual distance in kilometres.
7.6\times50000=380000 cm =3.8 km.
3. A flight leaves at 18:55 from a city at UTC+5 and lasts 7 h 40 min. Find the local arrival time in a city at UTC+1.
Departure in UTC is 13:55. Add 7 h 40 min to get 21:35 UTC. At UTC+1, local arrival is 22:35.
4. A cyclist travels 24 km at 16 km/h and returns along the same route at 24 km/h. Find the average speed.
Times are 1.5 h and 1 h. Total distance 48 km, total time 2.5 h, so average speed =19.2 km/h.
