Basic 8 · Mathematics · Decimals
Story Problems with Decimals and Laws of Indices
In this topic you will create and solve story problems involving decimals, and identify, explain and apply the laws of indices.
By the end of these notes you should be able to:
- Solve story problems that add and subtract decimals
- Solve story problems that multiply and divide decimals
- Identify and explain the laws of indices
What you already know
- Operations on decimals and powers
Copy everything that has a blue line on its left into your notebook. The other parts are for reading.
Lesson 1
- Strand
- Number
- Sub-strand
- Decimals
- Indicator
- B8.1.2.2.3 – Create and solve story problems involving decimals on the four basic operations
Introduction
Money, distances and masses often have decimals. In these problems, lining up the decimal point is everything.
1. Reminder
Line up decimal points, fill empty places with zeros and calculate as with whole numbers. Keep the decimal point in line in the answer.
2. Worked Problems
Shopping
Akua buys rice for GH₵84.50, fish for GH₵47.25 and oil for GH₵62.75. She pays with GH₵200. Find her change.
- Total: 84.50 + 47.25 + 62.75 = 194.50.
- Change: 200.00 − 194.50.
Answer: GH₵5.50
Journey
A cyclist rode 12.75 km before lunch and 8.6 km after lunch. How far did he ride, and how far is he from the end of a 30 km route?
- Distance ridden: 12.75 + 8.60 = 21.35 km.
- Remaining: 30.00 − 21.35 = 8.65 km.
Answer: 21.35 km ridden and 8.65 km left
3. Creating Story Problems
Good decimal problems come from real prices and measurements. Look at fuel prices, cloth lengths, body temperature, rainfall or running times. Write a clear question, and solve it yourself to check that the answer makes sense.
4. Rounding Answers
Money is given to 2 decimal places. Lengths may be given to 1 or 2 decimal places, depending on how they were measured. State the accuracy of your answer. Include the correct unit, such as GH₵, km, kg or litres.
5. Estimating with Decimals
Rounding to whole numbers gives a quick check. 84.50 + 47.25 + 62.75 is about 85 + 47 + 63 = 195, which agrees with the exact total of 194.50.
New words
- decimal
- A number with a decimal point.
- change
- Money returned after payment.
- unit
- A standard of measurement.
- accuracy
- How exact a value is.
Summary
- Line up decimal points when adding and subtracting.
- Story problems come from real prices and measurements.
- Include units and round to a sensible accuracy.
Class activity: Price Problems
- Collect prices from a shop or market.
- Create three story problems using decimals.
- Swap with another group and solve.
You need: Bottle tops, Ghana cedi play notes, Number cards
Exercises
Answer in your exercise book first, then check your answers.
1. 2.5 + 3.75 =
- A. 6.25
- B. 5.25
- C. 6.5
- D. 6.75
Show answer
Answer: 6.25Line up.2. 50 − 17.65 =
- A. 32.35
- B. 33.35
- C. 32.45
- D. 42.35
Show answer
Answer: 32.3550.00.3. Decimal points must be lined up when adding.
- A. True
- B. False
Show answer
Answer: TruePlace value.4. Money is usually given to ______ decimal places.
Show answer
Answer: 2Pesewas.5. Kofi ran 3.45 km on Monday and 4.8 km on Tuesday. Find the total.
Show answer
Answer: 8.25 kmAdd.6. A tank has 250.5 litres. 87.75 litres are used and 32.25 litres added. How much is in it?
Show answer
Answer: 195 litresTwo steps.
Lesson 2
- Strand
- Number
- Sub-strand
- Decimals
- Indicator
- B8.1.2.2.3 – Create and solve story problems involving decimals on the four basic operations
Introduction
If a litre of fuel costs GH₵14.50, what do 25 litres cost? These problems use multiplying and dividing decimals.
1. Reminder
To multiply decimals, multiply as whole numbers and count the total decimal places. To divide by a decimal, multiply both numbers by a power of ten so that the divisor is whole.
2. Worked Problems
Fuel cost
Find the cost of 25 litres of fuel at GH₵14.50 per litre.
- 14.50 × 25 = 14.50 × 100 ÷ 4.
- 1,450 ÷ 4 = 362.50.
Answer: GH₵362.50
Cutting cloth
A tailor has 22.5 m of cloth. Each dress needs 2.5 m. How many dresses can she make?
- 22.5 ÷ 2.5 = 225 ÷ 25.
Answer: 9 dresses
Unit price
6.5 kg of meat costs GH₵227.50. Find the price per kilogram.
- 227.50 ÷ 6.5 = 2,275 ÷ 65.
Answer: GH₵35 per kg
3. Unit Rates
A unit rate tells how much for one unit: price per kilogram, kilometres per litre, or cost per item. Comparing unit rates shows the better buy. If 2 kg of rice costs GH₵38 and 5 kg costs GH₵90, the unit rates are GH₵19 and GH₵18 per kg, so the 5 kg bag is cheaper per kilogram.
4. Mixed Problems
Some problems combine operations: a learner buys 3.5 m of cloth at GH₵22.40 per metre and pays with GH₵100. The cost is 3.5 × 22.40 = 78.40, and the change is 100 − 78.40 = GH₵21.60. Write each step clearly.
New words
- unit rate
- The amount for one unit.
- per
- For each.
- divisor
- The number you divide by.
- better buy
- The cheaper choice per unit.
Summary
- Count decimal places when multiplying, and make the divisor whole when dividing.
- Unit rates help to compare prices.
- Mixed problems combine several operations.
Class activity: Better Buy
- Compare the prices of two sizes of a product.
- Calculate the unit rate of each.
- Decide which is the better buy.
You need: Bottle tops, Ghana cedi play notes, Number cards
Exercises
Answer in your exercise book first, then check your answers.
1. 0.5 × 40 =
- A. 20
- B. 2
- C. 200
- D. 0.2
Show answer
Answer: 20Half of 40.2. 9.6 ÷ 0.4 =
- A. 24
- B. 2.4
- C. 240
- D. 0.24
Show answer
Answer: 2496 ÷ 4.3. A unit rate tells how much for one unit.
- A. True
- B. False
Show answer
Answer: TruePer unit.4. The word "per" means for ______.
Show answer
Answer: eachRate.5. Find the cost of 3.5 kg of beef at GH₵40 per kg.
Show answer
Answer: GH₵1403.5 × 40.6. 4 litres of oil cost GH₵92 and 5 litres cost GH₵110. Which is the better buy?
Show answer
Answer: 5 litres, at GH₵22 per litre against GH₵23.Unit rates.
Lesson 3
- Strand
- Number
- Sub-strand
- Indices
- Indicator
- B8.1.2.3.1 – Identify and explain the laws of indices and apply the laws of indices to simplify and evaluate numbers involving powers of numbers
Introduction
Powers follow simple laws. Once you know them, long multiplications with indices become very quick.
1. The Laws of Indices
| Law | Rule | Example |
|---|---|---|
| multiplication | aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ |
| division | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 5⁶ ÷ 5² = 5⁴ |
| power of a power | (aᵐ)ⁿ = aᵐⁿ | (3²)³ = 3⁶ |
| zero index | a⁰ = 1 | 7⁰ = 1 |
| negative index | a⁻ⁿ = 1/aⁿ | 2⁻³ = 1/8 |
| power of a product | (ab)ⁿ = aⁿbⁿ | (2 × 3)² = 2² × 3² |
2. Why the Laws Work
Write out the factors: 2³ × 2⁴ = (2 × 2 × 2) × (2 × 2 × 2 × 2), which is seven 2s, or 2⁷. So we add the indices. For division, 5⁶ ÷ 5² cancels two 5s, leaving 5⁴, so we subtract. For (3²)³, we have 3² three times: 3² × 3² × 3² = 3⁶, so we multiply.
3. Applying the Laws
Simplify
Simplify 3⁴ × 3⁵ ÷ 3⁶.
- Add the indices of the product: 3⁴⁺⁵ = 3⁹.
- Subtract for division: 3⁹⁻⁶ = 3³.
Answer: 27
4. Same Base Only
The first two laws work only when the bases are the same. You cannot simplify 2³ × 5⁴ by adding indices. You must evaluate each power: 8 × 625 = 5,000. Always check that the bases match. Also remember that 2³ × 2⁴ is 2⁷, not 4⁷: the base stays the same.
New words
- laws of indices
- Rules for working with powers.
- base
- The number being multiplied repeatedly.
- index
- The small number showing the power.
- negative index
- An index showing a reciprocal.
Summary
- When multiplying powers of the same base, add the indices; when dividing, subtract them.
- A power of a power multiplies the indices, and a⁰ = 1.
- A negative index means a reciprocal, and the laws need the same base.
Class activity: Law Discovery
- Write out 2³ × 2⁴ in full.
- Discover the rule for multiplying powers.
- Repeat for division and power of a power.
You need: Number cards, Square-grid paper, Calculator
Exercises
Answer in your exercise book first, then check your answers.
1. 2³ × 2² =
- A. 2⁵
- B. 2⁶
- C. 4⁵
- D. 4⁶
Show answer
Answer: 2⁵Add indices.2. (5²)³ =
- A. 5⁶
- B. 5⁵
- C. 5⁹
- D. 5⁸
Show answer
Answer: 5⁶Multiply indices.3. 7⁰ = 1.
- A. True
- B. False
Show answer
Answer: TrueZero index.4. 9⁵ ÷ 9³ = 9 to the power ______.
Show answer
Answer: 2Subtract.5. Simplify 4³ × 4⁵ ÷ 4⁶.
Show answer
Answer: 4² = 16Laws.6. Evaluate 2⁻³ and explain.
Show answer
Answer: 1/8, because a negative index means the reciprocal of 2³.Negative index.
Homework
- Solve five story problems with decimals.
- Write the laws of indices and give two examples of each.