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Basic 9 · Mathematics · Number and Numeration System

Significant Figures, Decimal Places and Place Value Problems

In this topic you will express numbers to a given number of significant figures and decimal places, and use place value to solve real-life problems.

By the end of these notes you should be able to:

  • Express numbers to given significant figures
  • Express numbers to given decimal places
  • Solve real-life place value problems

What you already know

  • Place value and rounding

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
Number and Numeration System
Indicator
B9.1.1.1.1 – Express integers to a given number of significant and decimal places

Introduction

A news report says that 33 million people live in Ghana. It does not give the exact count. Rounding gives us a number that is good enough.

1. What Are Significant Figures?

Significant figures: The digits in a number that carry meaning about its size, counted from the first non-zero digit.

Counting significant figures
NumberSignificant figuresReason
4,5724all digits count
0.00312leading zeros do not count
4,0024zeros between digits count
5.603final zero after a decimal point counts

2. Rounding to Significant Figures

  1. Count from the first non-zero digit.
  2. Look at the next digit.
  3. If it is 5 or more, round up.
  4. If it is less than 5, keep the digit.
  5. Fill any place holders with zeros.

Rounding

Express 3,486,721 to (a) 3 s.f. (b) 2 s.f.

  1. (a) The third digit is 8, and the next digit is 6, so round up: 3,490,000.
  2. (b) The second digit is 4, and the next is 8, so round up: 3,500,000.

Answer: 3,490,000 and 3,500,000

Small numbers

Express 0.004726 to 2 s.f.

  1. The first non-zero digit is 4.
  2. The second is 7, and the next is 2, so keep it.

Answer: 0.0047

A census worker with a tablet
Figure 1: Census worker Round five large numbers from a newspaper.

3. Why Round?

Exact figures may be unknown, or may give false accuracy. A census counts millions of people, and the figure changes every day. Reporting 33 million is honest and easy to remember. In science, the accuracy of an answer is limited by the measuring tool. If a length is measured to the nearest millimetre, there is no sense in giving a calculated answer to six decimal places.

4. Common Mistakes

Do not forget place holders. 3,486,721 to 3 s.f. is not 349: that changes the size of the number. Do not count the zeros at the start of a small number. And do not round in stages. Rounding 4,449 to 4,450 and then to 4,500 gives the wrong result. To 2 s.f., 4,449 is 4,400. Always go back to the original number.

New words

significant figure
A digit that carries meaning.
round
To replace a number with a simpler one.
place holder
A zero that keeps digits in position.
accuracy
How close a value is to the true value.

Summary

  • Count significant figures from the first non-zero digit.
  • Round using the next digit, and fill with zeros.
  • Always round from the original number.

Class activity: Rounding Newspaper

  1. Find large numbers in a newspaper.
  2. Round them to 2 and 3 s.f.
  3. Discuss when rounding is useful.

You need: Decimal place-value chart, Rounding-and-digit cards, Graph paper

Exercises

Answer in your exercise book first, then check your answers.

  1. 1. 0.0031 has how many s.f.?

    • A. 2
    • B. 4
    • C. 5
    • D. 3
    Show answer
    Answer: 2Leading zeros.

  2. 2. 45,678 to 3 s.f. is…

    • A. 45,700
    • B. 45,600
    • C. 457
    • D. 46,000
    Show answer
    Answer: 45,700Next digit 7.

  3. 3. Zeros between digits are significant.

    • A. True
    • B. False
    Show answer
    Answer: TrueRule.

  4. 4. 8,264 to 1 s.f. is ______.

    Show answer
    Answer: 8,000Rounding.

  5. 5. Express 0.05678 to 3 s.f.

    Show answer
    Answer: 0.0568Next digit 8.

  6. 6. Explain why 349 is a wrong answer for 3,486,721 to 3 s.f.

    Show answer
    Answer: It changes the size of the number, so zeros are needed: 3,490,000.Place holders.

Lesson 2

Strand
Number
Sub-strand
Number and Numeration System
Indicator
B9.1.1.1.1 – Express integers to a given number of significant and decimal places

Introduction

Prices have two decimal places, and scientific measurements may have more. How many we keep depends on the need.

1. What Are Decimal Places?

Decimal places: The digits to the right of the decimal point.

To round to a number of decimal places, count from the decimal point. Then look at the next digit. If it is 5 or more, add 1 to the last digit that you keep. Zeros at the end must be kept, as they show the accuracy of the answer.

Rounding 7.3964
AccuracyResultReason
3 d.p.7.396next digit 4
2 d.p.7.40next digit 6, so 9 rounds up
1 d.p.7.4next digit 9

Keep the zero

Express 12.5972 to 2 d.p.

  1. The second decimal digit is 9.
  2. The next digit is 7, so round up.
  3. 12.59 becomes 12.60.

Answer: 12.60

2. Significant Figures and Decimal Places

Comparing methods
Number2 s.f.2 d.p.
13.4871313.49
0.052380.0520.05
248.316250248.32

The two methods are different. Decimal places count from the point. Significant figures count from the first non-zero digit. Read the question carefully to see which is being asked for.

A fuel pump display with decimal numbers
Figure 2: Fuel pump display Round five calculator answers to 2 d.p.

3. Decimal Places with Money

Sharing a bill

GH₵100 is shared equally among 3 people. How much does each person get, to the nearest pesewa?

  1. 100 ÷ 3 = 33.3333...
  2. To 2 d.p. this is 33.33.

Answer: GH₵33.33

Note that 3 × 33.33 = 99.99, so a pesewa remains. In real life, one person must pay slightly more. Rounding can cause small differences like this, and accountants have to deal with them.

4. Using a Calculator

Calculators often show many digits. Do not copy them all. Round to the accuracy needed, and do so only at the end of a calculation, because early rounding can create larger errors. Write a slightly longer number in your working, then give your final rounded answer. Always state the accuracy you have used, for example "correct to 2 d.p."

New words

decimal place
A position after the decimal point.
pesewa
One hundredth of a cedi.
recurring
Repeating forever.
correct to
Rounded to a stated accuracy.

Summary

  • Count decimal places from the point.
  • Keep end zeros to show accuracy.
  • Round only at the end of a calculation.

Class activity: Calculator Rounding

  1. Use a calculator for five divisions.
  2. Round each answer to 1 and 2 d.p.
  3. Compare your results.

You need: Decimal place-value chart, Rounding-and-digit cards, Graph paper

Exercises

Answer in your exercise book first, then check your answers.

  1. 1. 4.567 to 1 d.p. is…

    • A. 4.6
    • B. 4.5
    • C. 4.57
    • D. 5
    Show answer
    Answer: 4.6Next digit 6.

  2. 2. 9.996 to 2 d.p. is…

    • A. 10.00
    • B. 9.99
    • C. 10.0
    • D. 9.90
    Show answer
    Answer: 10.00Rounds up.

  3. 3. End zeros can show accuracy.

    • A. True
    • B. False
    Show answer
    Answer: True7.40.

  4. 4. 3.14159 to 2 d.p. is ______.

    Show answer
    Answer: 3.14Next digit 1.

  5. 5. Express 0.08762 to 3 d.p.

    Show answer
    Answer: 0.088Next digit 6.

  6. 6. Why should you round only at the end?

    Show answer
    Answer: Early rounding can create larger errors.Accuracy.

Lesson 3

Strand
Number
Sub-strand
Number and Numeration System
Indicator
B9.1.1.1.2 – Use knowledge and understanding of place value to solve real life problems

Introduction

Budgets, populations and distances all use large numbers. Understanding place value helps you to make sense of them.

1. Reviewing Place Value

Place value in 5,207,348,190
DigitPlaceValue
5billions5,000,000,000
2hundred millions200,000,000
7millions7,000,000
3hundred thousands300,000
8thousands8,000
9tens90

2. Real-Life Problems

School budget

A district receives GH₵3,450,000 for 15 schools. If it is shared equally, how much does each school get?

  1. 3,450,000 ÷ 15 = 230,000.

Answer: GH₵230,000

Changing units

A company made a profit of GH₵2.4 million. Write this in full, and find the difference from GH₵1.75 million.

  1. 2.4 million = 2,400,000.
  2. 1.75 million = 1,750,000.
  3. 2,400,000 − 1,750,000 = 650,000.

Answer: GH₵2,400,000 and a difference of GH₵650,000

Ghanaian cedi banknotes
Figure 3: Ghana cedis Write three real-life numbers in words and figures.

3. Reading Large Numbers in the News

Newspapers often write "GH₵1.2 billion". This means 1.2 × 1,000,000,000 = 1,200,000,000. A million has six zeros, and a billion has nine zeros. Knowing how to change between these forms helps you to understand government budgets and business reports. It helps you to compare amounts: GH₵1.2 billion is 1,000 times GH₵1.2 million.

4. Estimating Reasonableness

Place value helps you to spot errors. If each of 30 learners pays GH₵15, the total must be about 30 × 15 = 450. An answer of 4,500 has a place value error. Before you accept an answer, estimate it. Round each number to one significant figure, and do the calculation in your head. This takes a few seconds, and it can save you several marks in an examination.

New words

billion
One thousand million.
budget
A plan for spending money.
profit
Money gained after costs.
estimate
A rough calculation.

Summary

  • Each digit's value depends on its place.
  • Large numbers can be written in words, figures or millions.
  • Estimate to catch place value errors.

Class activity: Budget Problems

  1. Find figures from a news report.
  2. Write them in full.
  3. Create two word problems.

You need: Place-value cards, Ghana cedi play notes, Graph paper

Exercises

Answer in your exercise book first, then check your answers.

  1. 1. A billion has how many zeros?

    • A. 9
    • B. 6
    • C. 12
    • D. 3
    Show answer
    Answer: 9Place value.

  2. 2. GH₵1.5 million is…

    • A. GH₵1,500,000
    • B. GH₵150,000
    • C. GH₵15,000,000
    • D. GH₵1,500
    Show answer
    Answer: GH₵1,500,000Convert.

  3. 3. Estimating can catch errors.

    • A. True
    • B. False
    Show answer
    Answer: TrueCheck.

  4. 4. In 7,458, the 4 is worth ______.

    Show answer
    Answer: 400Place value.

  5. 5. Write 3.2 billion in figures.

    Show answer
    Answer: 3,200,000,000Convert.

  6. 6. GH₵720,000 is shared among 9 towns. Find each town's share.

    Show answer
    Answer: GH₵80,000Divide.

Homework

  1. Solve five real-life place value problems from the news.
Significant Figures, Decimal Places and Place Value Problems – Basic 9 Mathematics Notes | sirbus