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Using hectares, Converting betweem kilometeres & miles

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visibility 123update 7 months agobookmarkshare

🎯 In this topic you will

Convert between hectares and square metres ($\text{m}^2$).
Convert between miles and kilometres.

🧠 Key Words

hectare (ha)
kilometre (km)
mile (mi)

Show Definitions

hectare (ha): A unit of area equal to $10{,}000 \,\text{m}^2$, commonly used for measuring land.
kilometre (km): A unit of length equal to $1000 \,\text{m}$, often used for measuring longer distances.
mile (mi): A unit of length equal to about $1609 \,\text{m}$, mainly used in the United States and United Kingdom.

You use hectares to measure areas of land.

A hectare is the area of a square field of side $100$ metres.

A football pitch is about half a hectare.

The abbreviation for hectare is ha.

You need to know this conversion:
$1 \text{ hectare (ha)} = 10000 \text{ m}^2$

📘 Worked example

a. Copy and complete these statements.

i. $2.4 \,\text{ha} = \_\_\_\_ \,\text{m}^2$ ii. $125000 \,\text{m}^2 = \_\_\_\_ \,\text{ha}$

b. A rectangular piece of land measures $850 \,\text{m}$ by $1.4 \,\text{km}$.
Work out the area of the land. Give your answer in hectares.

Answer:

a. i. $2.4 \times 10000 = 24000$
$2.4 \,\text{ha} = 24000 \,\text{m}^2$

a. ii. $125000 \div 10000 = 12.5$
$125000 \,\text{m}^2 = 12.5 \,\text{ha}$

b. $1.4 \,\text{km} = 1400 \,\text{m}$
Area $= 850 \times 1400$
$= 1190000 \,\text{m}^2$
$1190000 \div 10000 = 119 \,\text{ha}$

To convert hectares to square metres: Multiply the number of hectares by $10000$.

To convert square metres to hectares: Divide the number of square metres by $10000$.

For rectangular land problems: First convert all dimensions to metres, calculate the area in $\text{m}^2$, then divide by $10000$ to give the answer in hectares.

🧠 PROBLEM-SOLVING Strategy

Converting Between Units for Area and Distance

Follow these steps to decide when to multiply or divide by the conversion factor.

Identify the units given and the units required (e.g., from $\text{cm}^2$ to $\text{mm}^2$, or from km to miles).
Recall the correct conversion factor:
- $1 \,\text{cm}^2 = 100 \,\text{mm}^2$
- $1 \,\text{m}^2 = 10\,000 \,\text{cm}^2$
- $1 \,\text{ha} = 10\,000 \,\text{m}^2$
- $1 \,\text{km} \approx \tfrac{5}{8} \,\text{mile}$
- $1 \,\text{mile} \approx \tfrac{8}{5} \,\text{km}$
If converting to a smaller unit (e.g., m² → cm², km → miles), multiply by the conversion factor.
If converting to a larger unit (e.g., cm² → m², miles → km), divide by the conversion factor.
Always check if the answer is reasonable by estimating (e.g., a hectare is very large, so the number in m² should be much bigger).
For compound word problems:
- First, convert all lengths to the same unit.
- Find the area using the correct formula (e.g., rectangle: $A = l \times w$).
- Convert the area into the required unit (m², ha, km²).
- Apply costs, rates, or ratios as needed.

❓ EXERCISES

1. Copy and complete these conversions between hectares and m$^2$.

a. $6 \,\text{ha} = \_\_\_\_ \,\text{m}^2$

b. $11.2 \,\text{ha} = \_\_\_\_ \,\text{m}^2$

c. $0.63 \,\text{ha} = \_\_\_\_ \,\text{m}^2$

👀 Show answer

a. $60000 \,\text{m}^2$
b. $112000 \,\text{m}^2$
c. $6300 \,\text{m}^2$

2. Copy and complete these conversions.

a. $4.6 \,\text{ha} = \_\_\_\_ \,\text{m}^2$

b. $0.8 \,\text{ha} = \_\_\_\_ \,\text{m}^2$

c. $0.75 \,\text{ha} = \_\_\_\_ \,\text{m}^2$

d. $0.025 \,\text{ha} = \_\_\_\_ \,\text{m}^2$

👀 Show answer

a. $46000 \,\text{m}^2$
b. $8000 \,\text{m}^2$
c. $7500 \,\text{m}^2$
d. $250 \,\text{m}^2$

3. Copy and complete these conversions between m$^2$ and hectares.

a. $70000 \,\text{m}^2 = \_\_\_\_ \,\text{ha}$

b. $135000 \,\text{m}^2 = \_\_\_\_ \,\text{ha}$

c. $8000 \,\text{m}^2 = \_\_\_\_ \,\text{ha}$

👀 Show answer

a. $7 \,\text{ha}$
b. $13.5 \,\text{ha}$
c. $0.8 \,\text{ha}$

4. Copy and complete these conversions.

a. $89000 \,\text{m}^2 = \_\_\_\_ \,\text{ha}$

b. $240000 \,\text{m}^2 = \_\_\_\_ \,\text{ha}$

c. $900 \,\text{m}^2 = \_\_\_\_ \,\text{ha}$

d. $1265000 \,\text{m}^2 = \_\_\_\_ \,\text{ha}$

👀 Show answer

a. $8.9 \,\text{ha}$
b. $24 \,\text{ha}$
c. $0.09 \,\text{ha}$
d. $126.5 \,\text{ha}$

💡 Tip

In questions 5 to 7, use the formula for the area of a rectangle. See Section 2.2 for a reminder on using formulae.

5. A rectangular piece of land measures $780 \,\text{m}$ by $550 \,\text{m}$. Work out the area of the land, in:

a. square metres

b. hectares

👀 Show answer

a. $780 \times 550 = 429000 \,\text{m}^2$
b. $429000 \div 10000 = 42.9 \,\text{ha}$

6. A builder buys a rectangular piece of land. The dimensions of the land are shown in the diagram.

a. Work out the area of the land, in hectares.

The cost of the land is $12400$ per hectare. The builder says, “This land will cost me more than $34000$.”

b. Is the builder correct? Explain your answer. Show your working.

👀 Show answer

a. Area $= 350 \times 80 = 28000 \,\text{m}^2 = 2.8 \,\text{ha}$
b. Cost $= 2.8 \times 12400 = 34720$
Yes, the builder is correct (the cost is more than $34000$).

7. A football pitch has an area of $0.78 \,\text{ha}$.

a. Work out the area of the football pitch, in m$^2$.

b. The length of the football pitch is $120 \,\text{m}$. Work out the width of the football pitch, in metres.

👀 Show answer

a. $0.78 \times 10000 = 7800 \,\text{m}^2$
b. Width $= 7800 \div 120 = 65 \,\text{m}$

8. Alessia and Ben share a piece of land in the ratio $1:2$. The area of the piece of land they share is $0.087 \,\text{hectares}$. Work out the area of Ben’s piece, in m$^2$.

👀 Show answer

Total area $= 0.087 \times 10000 = 870 \,\text{m}^2$
Ratio $1:2 \implies \text{Ben’s share} = \tfrac{2}{3} \times 870 = 580 \,\text{m}^2$

🧠 Think like a Mathematician

9. Look back at Question 8. Reflect on the different methods you could use to answer the question.

a. What are the advantages and disadvantages of the different methods?

b. Which method do you think is the best? Explain why.

👀 show answer

a. - One method may involve first converting hectares to square metres before applying the ratio. - Another method may work directly in hectares and then convert the result to m$^2$. - Converting early can make the arithmetic easier to follow, but may involve larger numbers. - Working in hectares avoids big numbers but requires a final conversion step.
b. The best method is the one that keeps the arithmetic simplest and avoids unnecessary conversions. Often, working directly in hectares and then converting the final answer is clearer and less error-prone.

❓ EXERCISES

💡 Tip

Start by changing the measurements given on the plan to metres.

10. A company wants to build a water park. The diagram shows a plan of the land the company wants to buy.

The price of the land is $5200$ per hectare. The company wants to pay less than $5$ million for the land.

Can the company afford to buy the land? Show all your working. Explain your answer.

👀 Show answer

Convert dimensions to metres:
$2 \,\text{km} = 2000 \,\text{m}$, $3 \,\text{km} = 3000 \,\text{m}$, $4 \,\text{km} = 4000 \,\text{m}$, $1.5 \,\text{km} = 1500 \,\text{m}$.

Divide the shape into two rectangles:
- Rectangle A: $2000 \times 2500 = 5000000 \,\text{m}^2$
- Rectangle B: $3000 \times 1500 = 4500000 \,\text{m}^2$
Total area $= 9500000 \,\text{m}^2 = 950 \,\text{ha}$.

Cost $= 950 \times 5200 = 4940000$.

Since $4940000 < 5000000$, the company can afford the land.

In some countries, such as the USA, Liberia and the UK, distances are measured in miles rather than kilometres.

A kilometre is a shorter unit of measurement than a mile.

One kilometre is about $\tfrac{5}{8}$ of a mile.

If the blue line below represents a distance of 1 mile, then the red line represents a distance of 1 kilometre.

To convert a distance in kilometres to a distance in miles, multiply by $\tfrac{5}{8}$.

To convert a distance in miles to a distance in kilometres, multiply by $\tfrac{8}{5}$.

📘 Worked example

a. Which is greater, 20 miles or 20 km?

b. Convert 72 kilometres into miles.

c. Convert 50 miles into kilometres.

d. Which is further, 200 km or 120 miles?

Answer:

a. 20 miles.
1 mile is greater than 1 km, so 20 miles is greater than 20 km.

b. $72 \div 8 = 9$
$9 \times 5 = 45 \,\text{miles}$

c. $50 \div 5 = 10$
$10 \times 8 = 80 \,\text{km}$

d. $200 \div 8 = 25$
$25 \times 5 = 125 \,\text{miles}$
200 km is further (125 miles is greater than 120 miles).

For part b: To multiply 72 by $\tfrac{5}{8}$, first divide 72 by 8, then multiply the answer by 5.

For part c: To multiply 50 by $\tfrac{8}{5}$, first divide 50 by 5, then multiply the answer by 8.

For part d: Convert 200 km into miles (or 120 miles into km) so the units are the same. Since 125 miles is greater than 120 miles, 200 km is further.

❓ EXERCISES

1. Write true (T) or false (F) for each statement.

a. 15 miles is further than 15 km.

b. 100 km is exactly the same distance as 100 miles.

c. 2.5 km is further than 2.5 miles.

d. 6 km is not as far as 6 miles.

e. In one hour, a car travelling at 70 miles per hour will travel a shorter distance than a car travelling at 70 kilometres per hour.

👀 Show answer

a. True (since 1 mile > 1 km).
b. False (100 km ≈ 62 miles).
c. False (2.5 miles ≈ 4 km, so it’s further).
d. True.
e. False (70 mph ≈ 112 km/h, which is further than 70 km/h).

2. Read what Zara says.
“My brother lives 35 km from my house. My sister lives 35 miles from my house. I live closer to my brother than to my sister.”
Is Zara correct? Explain your answer.

👀 Show answer

35 miles ≈ 56 km, which is further than 35 km. So yes, Zara is correct that her brother is closer.

3. Copy and complete these conversions from kilometres to miles.

a. 64 km = $64 \div 8 = 8$ ; $8 \times 5 = \_\_\_$ miles

b. 40 km = $40 \div 8 = \_\_$ ; $\_\_ \times 5 = \_\_\_$ miles

c. 56 km = $56 \div \_\_ = \_\_$ ; $\_\_ \times 5 = \_\_\_$ miles

👀 Show answer

a. $64 \div 8 = 8$, $8 \times 5 = 40$ miles.
b. $40 \div 8 = 5$, $5 \times 5 = 25$ miles.
c. $56 \div 8 = 7$, $7 \times 5 = 35$ miles.

4. Copy and complete these conversions from miles to kilometres.

a. 55 miles = $55 \div 5 = 11$ ; $11 \times 8 = \_\_\_$ km

b. 20 miles = $20 \div 5 = \_\_$ ; $\_\_ \times 8 = \_\_\_$ km

c. 85 miles = $85 \div \_\_ = \_\_$ ; $\_\_ \times 8 = \_\_\_$ km

👀 Show answer

a. $55 \div 5 = 11$, $11 \times 8 = 88$ km.
b. $20 \div 5 = 4$, $4 \times 8 = 32$ km.
c. $85 \div 5 = 17$, $17 \times 8 = 136$ km.

🧠 Think like a Mathematician

5. Read what Sofia says.
Discuss a strategy Sofia could use to help her decide when she should multiply by $\tfrac{5}{8}$ and when she should multiply by $\tfrac{8}{5}$.

👀 show answer

If the question gives a distance in kilometres and asks for miles, multiply by $\tfrac{5}{8}$.
If the question gives a distance in miles and asks for kilometres, multiply by $\tfrac{8}{5}$.
A good strategy is to write down what units you are starting with and what units you need to end with, then apply the correct conversion factor.

❓ EXERCISES

6. Convert each distance to miles.

a. $24 \,\text{km}$

b. $48 \,\text{km}$

c. $96 \,\text{km}$

d. $176 \,\text{km}$

👀 Show answer

a. $24 \div 8 = 3$, $3 \times 5 = 15 \,\text{miles}$
b. $48 \div 8 = 6$, $6 \times 5 = 30 \,\text{miles}$
c. $96 \div 8 = 12$, $12 \times 5 = 60 \,\text{miles}$
d. $176 \div 8 = 22$, $22 \times 5 = 110 \,\text{miles}$

7. Convert each distance to kilometres.

a. $10 \,\text{miles}$

b. $100 \,\text{miles}$

c. $125 \,\text{miles}$

d. $180 \,\text{miles}$

👀 Show answer

a. $10 \div 5 = 2$, $2 \times 8 = 16 \,\text{km}$
b. $100 \div 5 = 20$, $20 \times 8 = 160 \,\text{km}$
c. $125 \div 5 = 25$, $25 \times 8 = 200 \,\text{km}$
d. $180 \div 5 = 36$, $36 \times 8 = 288 \,\text{km}$

🧠 Think like a Mathematician

8. Look at this question:

Which is further, 107 km or 70 miles?

Reflect on the following:

a. Do you think it is easier to change 107 km into miles or 70 miles into km without using a calculator? Explain why.

b. If you could use a calculator, would this change your answer to part a? Explain why.

c. When you compare a number of km and a number of miles, explain how you would decide which unit to convert.

d. Test your answer to part c on these questions:

i. Which is further, 90 miles or 150 km?
ii. Which is further, 51 miles or 80 km?

👀 show answer

a: It is easier to convert 70 miles into km, because multiplying by $\tfrac{8}{5}$ is straightforward.
b: With a calculator, either direction works. The result is the same, so convenience matters less.
c: Choose to convert into the unit that makes the calculation simplest, usually by multiplying rather than dividing.
d(i): $90 \times \tfrac{8}{5} = 144 \,\text{km}$, which is less than 150 km → 150 km is further.
d(ii): $51 \times \tfrac{8}{5} = 81.6 \,\text{km}$, which is greater than 80 km → 51 miles is further.

❓ EXERCISES

9. Use only the numbers from the cloud to complete these statements.

a. $120 \,\text{km} = \_\_ \,\text{miles}$

b. $105 \,\text{miles} = \_\_ \,\text{km}$

c. $\_\_ \,\text{km} = \_\_ \,\text{miles}$

d. $\_\_ \,\text{miles} = \_\_ \,\text{km}$

👀 Show answer

a. $120 \div 8 = 15$, $15 \times 5 = 75 \,\text{miles}$
b. $105 \div 5 = 21$, $21 \times 8 = 168 \,\text{km}$
c. $224 \,\text{km} = 140 \,\text{miles}$
d. $115 \,\text{miles} = 184 \,\text{km}$

❓ EXERCISES

💡 Tip

Give each answer as a mixed number in its simplest form.

10. Work out the missing numbers in these conversions. Use your preferred method.

a. $17 \,\text{miles} = \_\_ \,\text{km}$

b. $33 \,\text{miles} = \_\_ \,\text{km}$

c. $54 \,\text{miles} = \_\_ \,\text{km}$

d. $28 \,\text{km} = \_\_ \,\text{miles}$

e. $42 \,\text{km} = \_\_ \,\text{miles}$

f. $75 \,\text{km} = \_\_ \,\text{miles}$

👀 Show answer

a. $17 \times \tfrac{8}{5} = 27.2 \,\text{km}$
b. $33 \times \tfrac{8}{5} = 52.8 \,\text{km}$
c. $54 \times \tfrac{8}{5} = 86.4 \,\text{km}$
d. $28 \times \tfrac{5}{8} = 17.5 \,\text{miles}$
e. $42 \times \tfrac{5}{8} = 26.25 \,\text{miles}$
f. $75 \times \tfrac{5}{8} = 46.875 \,\text{miles}$

11. Every car in the USA is fitted with a milometer. The milometer shows the total distance a car has travelled. Evan is a salesman.

This is the reading on his car’s milometer at the start of one week.
$125\,465$ miles

This is the reading on his car’s milometer at the end of the week.
$126\,335$ miles

a. How many kilometres has Evan travelled in this week?

b. Evan is paid 20 cents for each kilometre he travels. This is to pay for the fuel he uses. Evan works out that, in this week, he will be paid more than \$250 for the fuel he uses. Is Evan correct? Explain your answer.

👀 Show answer

a. Distance travelled = $126335 - 125465 = 870 \,\text{miles}$.
Convert: $870 \times \tfrac{8}{5} = 1392 \,\text{km}$.

b. Payment = $1392 \times 0.20 = 278.4 \,\$.$
Since \$278.40 is greater than \$250, Evan is correct.

📘 What we've learned

A hectare is a unit of area equal to $10\,000 \,\text{m}^2$, often used to measure land.
To convert from hectares to square metres, multiply by $10\,000$; to convert back, divide by $10\,000$.
We practiced finding areas of rectangles in metres, then expressing them in hectares.
1 kilometre is about $\tfrac{5}{8}$ of a mile, and 1 mile is about $\tfrac{8}{5}$ of a kilometre.
To convert from kilometres to miles, multiply by $\tfrac{5}{8}$.
To convert from miles to kilometres, multiply by $\tfrac{8}{5}$.
We compared distances expressed in different units and applied conversions in real-world problems (e.g., land cost, travel distances).

Using hectares, Converting betweem kilometeres & miles

🎯 In this topic you will

🧠 Key Words

🧠 PROBLEM-SOLVING Strategy

Converting Between Units for Area and Distance

❓ EXERCISES

💡 Tip

🧠 Think like a Mathematician

❓ EXERCISES

💡 Tip

❓ EXERCISES

🧠 Think like a Mathematician

❓ EXERCISES

🧠 Think like a Mathematician

❓ EXERCISES

❓ EXERCISES

💡 Tip

📘 What we've learned

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