Basic 7 · Mathematics · Numeration Systems
Decimal Places, Significant Figures and Powers of Ten
In this topic you will correct numbers to a given number of decimal places and significant figures, and multiply and divide whole numbers and decimals by powers of 10 and by 0.1, 0.01 and other benchmark fractions of 10.
By the end of these notes you should be able to:
- Correct numbers to a given number of decimal places
- Correct numbers to a given number of significant figures
- Multiply and divide numbers by powers of 10 and by decimal fractions of 10
What you already know
- Rounding whole numbers and decimals
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
- Numeration Systems
- Indicator
- B7.1.1.1.5 – Express decimal numerals to given significant and decimal places
Introduction
Scientists, engineers and shopkeepers often need answers that are accurate to a certain number of decimal places. "Correct to 2 decimal places" means keep exactly two digits after the decimal point.
1. What Are Decimal Places?
Decimal places: The number of digits after the decimal point. Written as d.p.
4.7 has one decimal place; 4.73 has two; 4.738 has three.
2. Correcting to a Given Number of Decimal Places
- Count the required number of digits after the decimal point.
- Look at the next digit.
- If it is 5 or more, add 1 to the last digit kept.
- If it is less than 5, leave the last digit unchanged.
- Drop all the digits after it.
Correct to 2 d.p.
Correct 15.6387 to 2 decimal places.
- Keep two digits after the point: 15.63.
- Next digit: 8, which is 5 or more.
- Add 1 to 3.
Answer: 15.64
Correct to 1 d.p.
Correct 0.0492 to 1 decimal place.
- Keep one digit after the point: 0.0.
- Next digit: 4, which is less than 5.
Answer: 0.0
3. Decimal Places in Calculations
When you divide, the answer may have many decimal places. For example, 22 ÷ 7 = 3.142857... Correct to 2 d.p., this is 3.14; to 3 d.p., it is 3.143. Always state the number of decimal places you have used.
4. Why Decimal Places Matter
Different situations need different accuracy. Money needs 2 decimal places because the smallest unit is a pesewa. A carpenter may measure to 1 decimal place of a centimetre. A scientist measuring medicine may need 3 or more decimal places, because a small error could be dangerous. Choosing the right number of decimal places makes an answer both accurate and practical.
New words
- decimal places
- Digits after the decimal point.
- d.p.
- Short form of decimal places.
- correct to
- Rounded to a stated accuracy.
- recurring decimal
- A decimal in which digits repeat forever.
Summary
- Decimal places are the digits after the decimal point.
- To correct to n d.p., keep n digits and look at the next digit to decide whether to round up.
- State the number of decimal places used in an answer.
Class activity: Decimal Place Practice
- Use a calculator to find 10 ÷ 3, 5 ÷ 7 and 22 ÷ 7.
- Correct each answer to 1, 2 and 3 decimal places.
- Compare your answers with your partner's.
You need: Decimal place-value chart, Digit-and-rounding cards, Graph paper
Exercises
Answer in your exercise book first, then check your answers.
1. How many decimal places does 3.456 have?
- A. 3
- B. 2
- C. 4
- D. 1
Show answer
Answer: 3Three digits after the point.2. 12.3456 correct to 2 d.p. is…
- A. 12.35
- B. 12.34
- C. 12.3
- D. 12.346
Show answer
Answer: 12.35Next digit 5.3. 7.98 correct to 1 d.p. is 8.0.
- A. True
- B. False
Show answer
Answer: TrueRounding up 9 gives 8.0.4. The short form of decimal places is ______.
Show answer
Answer: d.p.e.g. 2 d.p.5. Correct 0.07654 to 3 decimal places.
Show answer
Answer: 0.077Next digit 5.6. Correct 22 ÷ 7 to 3 decimal places and explain your steps.
Show answer
Answer: 3.143; 22 ÷ 7 = 3.142857...; keep 3.142, next digit 8, so round up.Division then rounding.
Lesson 2
- Strand
- Number
- Sub-strand
- Numeration Systems
- Indicator
- B7.1.1.1.5 – Express decimal numerals to given significant and decimal places
Introduction
When measuring the distance from Accra to Kumasi, we might say about 250 km. The "2" and "5" are the important digits. These important digits are called significant figures.
1. Rules for Significant Figures
| Rule | Example | Significant figures |
|---|---|---|
| all non-zero digits are significant | 347 | 3 |
| zeros between non-zero digits are significant | 4,005 | 4 |
| leading zeros (at the start) are not significant | 0.0034 | 2 |
| trailing zeros after a decimal point are significant | 2.50 | 3 |
| trailing zeros in a whole number may or may not be significant | 5,000 | 1 (unless stated) |
2. Correcting to Significant Figures
- Start counting from the first non-zero digit.
- Count the required number of significant figures.
- Look at the next digit and round up or down.
- For whole numbers, replace dropped digits with zeros to keep the place value.
Correct to 3 s.f.
Correct 45,862 to 3 significant figures.
- First three significant figures: 4, 5, 8.
- Next digit: 6, so round up 8 to 9.
- Replace the remaining digits with zeros.
Answer: 45,900
Correct to 2 s.f.
Correct 0.004718 to 2 significant figures.
- First non-zero digit: 4. First two significant figures: 4, 7.
- Next digit: 1, so round down.
Answer: 0.0047
3. Decimal Places and Significant Figures Compared
Decimal places count digits after the point; significant figures count digits from the first non-zero digit. For 0.06538: to 2 d.p. it is 0.07, but to 2 s.f. it is 0.065. Read questions carefully to know which one is asked.
New words
- significant figures
- The digits that carry meaning in a number, counted from the first non-zero digit.
- s.f.
- Short form of significant figures.
- leading zero
- A zero before the first non-zero digit.
- accuracy
- How close a value is to the true value.
Summary
- Significant figures are counted from the first non-zero digit.
- Leading zeros are not significant; zeros between digits are.
- For whole numbers, replace dropped digits with zeros when correcting to s.f.
Class activity: Significant Figure Sort
- Sort number cards by how many significant figures they have.
- Correct each number to 2 s.f. and 3 s.f.
- Explain one tricky example to the class.
You need: Decimal place-value chart, Digit-and-rounding cards, Graph paper
Exercises
Answer in your exercise book first, then check your answers.
1. How many significant figures does 2,805 have?
- A. 4
- B. 3
- C. 2
- D. 1
Show answer
Answer: 4Zero between digits counts.2. 0.08362 correct to 2 s.f. is…
- A. 0.084
- B. 0.08
- C. 0.083
- D. 0.0836
Show answer
Answer: 0.084Next digit 6.3. Leading zeros are significant.
- A. True
- B. False
Show answer
Answer: FalseThey only show place.4. The short form of significant figures is ______.
Show answer
Answer: s.f.e.g. 3 s.f.5. Correct 763,429 to 2 significant figures.
Show answer
Answer: 760,000Next digit 3.6. Correct 0.06538 to 2 d.p. and to 2 s.f., and explain the difference.
Show answer
Answer: 0.07 (2 d.p.) and 0.065 (2 s.f.); d.p. counts after the point, s.f. counts from the first non-zero digit.Different counts.
Lesson 3
- Strand
- Number
- Sub-strand
- Number Operations
- Indicator
- B7.1.2.1.1 – Multiply and divide given numbers by powers of 10 including decimals and benchmark fractions of 10
Introduction
Multiplying by 10, 100 or 1,000 is one of the quickest calculations in mathematics. Each time we multiply by 10, every digit moves one place to the left.
1. Multiplying by 10, 100 and 1,000
| Calculation | Answer | Movement |
|---|---|---|
| 3.45 × 10 | 34.5 | digits move 1 place left |
| 3.45 × 100 | 345 | 2 places left |
| 3.45 × 1,000 | 3,450 | 3 places left |
| 27 × 10⁴ | 270,000 | 4 places left |
2. Dividing by 10, 100 and 1,000
| Calculation | Answer | Movement |
|---|---|---|
| 562 ÷ 10 | 56.2 | digits move 1 place right |
| 562 ÷ 100 | 5.62 | 2 places right |
| 562 ÷ 1,000 | 0.562 | 3 places right |
| 4.8 ÷ 100 | 0.048 | 2 places right |
3. Multiplying and Dividing by 0.1, 0.01 and 1/10
Multiplying by 0.1 or 1/10 is the same as dividing by 10. Multiplying by 0.01 or 1/100 is the same as dividing by 100. Dividing by 0.1 is the same as multiplying by 10.
Calculate
Find 45 × 0.1, 45 × 0.01 and 45 ÷ 0.1.
- 45 × 0.1 = 45 ÷ 10 = 4.5
- 45 × 0.01 = 45 ÷ 100 = 0.45
- 45 ÷ 0.1 = 45 × 10 = 450
Answer: 4.5, 0.45 and 450
4. Real-Life Use
Converting units uses powers of ten. 1 metre = 100 centimetres, so 2.35 m = 2.35 × 100 = 235 cm. 1 kilogram = 1,000 grams, so 4,500 g = 4,500 ÷ 1,000 = 4.5 kg. 1 cedi = 100 pesewas, so 350 pesewas = 350 ÷ 100 = GH₵3.50.
New words
- power of ten
- 10, 100, 1,000 and so on, written 10¹, 10², 10³.
- place-value shift
- The movement of digits when multiplying or dividing by powers of ten.
- benchmark fraction
- A common fraction such as 1/10 or 1/100.
- convert
- To change from one unit to another.
Summary
- Multiplying by 10ⁿ moves digits n places left; dividing moves them n places right.
- Multiplying by 0.1 or 1/10 is the same as dividing by 10.
- Powers of ten are used to convert units such as metres, grams and cedis.
Class activity: Unit Conversion Station
- Measure three objects in metres and convert to centimetres.
- Weigh three items in grams and convert to kilograms.
- Record the calculations using powers of ten.
You need: Fraction strips, Bottle tops, Graph paper
Exercises
Answer in your exercise book first, then check your answers.
1. 6.7 × 100 =
- A. 670
- B. 67
- C. 6,700
- D. 0.67
Show answer
Answer: 670Two places left.2. 38 × 0.01 =
- A. 0.38
- B. 3.8
- C. 380
- D. 0.038
Show answer
Answer: 0.38Same as ÷ 100.3. Dividing by 0.1 is the same as multiplying by 10.
- A. True
- B. False
Show answer
Answer: TrueInverse.4. 2.5 kg = ______ g.
Show answer
Answer: 2,500× 1,000.5. Calculate 0.72 ÷ 100 and 0.72 × 1,000.
Show answer
Answer: 0.0072 and 720Place-value shifts.6. Convert 1,850 pesewas to cedis and explain.
Show answer
Answer: GH₵18.50, because 1,850 ÷ 100 = 18.5.100 pesewas = 1 cedi.
Homework
- Correct five measurements from home to 1 d.p. and 2 s.f.
- Convert five lengths from metres to centimetres and five masses from grams to kilograms.