Decimals and converting fractions, decimals and percentages

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Notes de leçon

Place Value in Decimals

Each digit in a decimal number has a place value that is a power of 10.
The decimal point separates the whole-number part from the fractional part.
To the right of the decimal point, the place values are tenths, hundredths, thousandths, and so on: 0.1, 0.01, 0.001, …
For example, in 3.456, the 4 is in the tenths place, the 5 in the hundredths place, and the 6 in the thousandths place.
The value of a digit is the digit multiplied by its place value: in 3.456, the 4 represents 4 × 0.1 = 0.4.
Zeros can be added to the right of a decimal without changing its value: 5.2 = 5.20 = 5.200.
Zeros on the left of a number do not change its value: 3.14 = 03.14 = 003.14.

Multiplying and Dividing by Powers of 10

Multiplying by 10 moves each digit one place to the left, so the decimal point moves one place to the right.
Dividing by 10 moves each digit one place to the right, so the decimal point moves one place to the left.
For example, 3.45 × 10 = 34.5 and 3.45 ÷ 10 = 0.345.
Multiplying by 100 moves the decimal point two places to the right; dividing by 100 moves it two places to the left.
In general, multiplying by 10ⁿ moves the decimal point n places to the right; dividing by 10ⁿ moves it n places to the left.
If there are not enough digits, add zeros as placeholders: 2.5 × 100 = 250, 0.3 ÷ 100 = 0.003.

Calculating with Decimals

To add or subtract decimals, line up the decimal points and then add or subtract as usual.
Write zeros so that both numbers have the same number of decimal places: 3.5 + 2.75 becomes 3.50 + 2.75.
To multiply decimals, ignore the decimal points and multiply the whole numbers, then put the decimal point back so that the answer has the same total number of decimal places as the two numbers combined.
For example, 0.3 × 0.4: 3 × 4 = 12, and there are two decimal places in total, so the answer is 0.12.
To divide by a decimal, multiply both numbers by a power of 10 to make the divisor a whole number: 4.8 ÷ 0.6 = 48 ÷ 6 = 8.
When solving problems, check that your answer is reasonable by estimating first.

Converting Fractions to Decimals

To convert a fraction to a decimal, divide the numerator by the denominator.
For example, 3/4 = 3 ÷ 4 = 0.75.
Some fractions give terminating decimals (they stop after a finite number of digits), such as 1/2 = 0.5, 1/4 = 0.25, 1/5 = 0.2.
Other fractions give recurring decimals (they repeat a pattern forever), such as 1/3 = 0.333… and 2/11 = 0.181818…
A fraction in simplest form gives a terminating decimal if and only if its denominator has only 2 and 5 as prime factors.
For example, 3/8 terminates because 8 = 2³; 1/6 recurs because 6 = 2 × 3.

Converting Decimals to Fractions

To convert a terminating decimal to a fraction, use the place value of the last digit.
For example, 0.7 = 7/10, 0.45 = 45/100 = 9/20, 0.125 = 125/1000 = 1/8.
Always simplify the fraction to its lowest terms.
To convert a recurring decimal to a fraction, use a method involving multiplying by powers of 10 and subtracting.
For example, let x = 0.777… Then 10x = 7.777…, so 9x = 7, giving x = 7/9.
Another example: 0.363636… = 36/99 = 4/11.

Recurring Decimals and Dot Notation

A recurring decimal has a digit or block of digits that repeats forever.
Dot notation is used to show which digits recur: a dot is placed over the first and last digit of the repeating block.
For example, 0.333… is written as 0.̇3 (dot over the 3).
0.181818… is written as 0.̇1̇8 (dots over the 1 and 8).
If only one digit recurs, one dot is used; if more than one digit recurs, dots are placed over the first and last digits of the repeating sequence.
Every recurring decimal represents a rational number (a fraction).

Converting Between Fractions, Decimals and Percentages

To convert a decimal to a percentage, multiply by 100 and add the % sign: 0.75 = 75%.
To convert a percentage to a decimal, divide by 100: 45% = 0.45.
To convert a fraction to a percentage, first convert to a decimal, then multiply by 100: 3/5 = 0.6 = 60%.
To convert a percentage to a fraction, write it as a fraction with denominator 100 and simplify: 40% = 40/100 = 2/5.
Common equivalents to learn: 1/2 = 0.5 = 50%, 1/4 = 0.25 = 25%, 3/4 = 0.75 = 75%, 1/5 = 0.2 = 20%, 1/10 = 0.1 = 10%.
Also: 1/3 ≈ 0.333… ≈ 33.3%, 2/3 ≈ 0.666… ≈ 66.7%, 1/8 = 0.125 = 12.5%.

Ordering Fractions, Decimals and Percentages

To compare and order a mixed list, convert all numbers to the same form, usually decimals.
For example, to order 0.6, 3/5, 55%, convert: 0.6 = 0.6, 3/5 = 0.6, 55% = 0.55.
Then compare the decimal values: 0.55 < 0.6 = 0.6, so 55% < 0.6 = 3/5.
When ordering, be careful with recurring decimals: 0.666… is greater than 0.6 but less than 0.67.
Use the symbols < (less than), > (greater than), and = (equal to) to show the order.
Always double-check by converting back to the original form if needed.

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1.What is the value of the digit 7 in the number 3.274?
Easy
- A7 tenths
- B7 hundredths
- C7 thousandths
- D7 units
2.Which of these is the correct result of multiplying 0.45 by 100?
Easy
- A0.0045
- B4.5
- C45
- D450
3.Which fraction is equivalent to 0.375?
Medium
- A\frac{1}{8}
- B\frac{3}{8}
- C\frac{3}{4}
- D\frac{5}{8}
4.Which of the following decimals are terminating? (select all that apply)
Medium
- A0.25
- B0.333...
- C0.125
- D0.666...
- E0.1
5.The decimal 0.333... is a recurring decimal.
Easy
True or false?
6.Match each fraction to its equivalent percentage.
Medium
- 1/2
- 1/5
- 3/4
- 2/3
- 50%
- 20%
- 75%
- 66.6...%
7.Order these numbers from smallest to largest: 0.6, 65%, 3/5, 0.66.
Medium
- 0.6
- 65%
- 3/5
- 0.66
8.How many decimal places does 4.690 have?
Medium
- A1
- B2
- C3
- D4

Plus de questions

9.The trailing zero in 4.690 changes the value of the number.
Easy
True or false?
10.In base-ten, the digit 0 is a decimal digit.
Easy
True or false?
11.Which number is written correctly with a decimal separator?
Easy
- A25.9703
- B25,9703
- C25 9703
- D25.9703.5
12.Match each decimal to its meaning in words.
Medium
- 0.7
- 0.07
- 0.007
- 7.0
- seven tenths
- seven hundredths
- seven thousandths
- seven units
13.Order these decimals from smallest to largest.
Medium
- 0.1
- 0.01
- 0.001
- 0.0001
14.Which of the following fractions is equivalent to the recurring decimal 0.444...?
Medium
- A4/9
- B4/10
- C44/100
- D4/11
15.Which of the following is the correct decimal expansion of 1/3 to three decimal places?
Medium
- A0.333
- B0.334
- C0.33
- D0.3333
16.What is 4.7 × 100?
Easy
- A0.047
- B47
- C470
- D4700
17.What is 3.6 ÷ 10?
Medium
- A0.036
- B0.36
- C36
- D360
18.What is 5.2 + 0.78?
Medium
- A5.98
- B6.0
- C5.80
- D5.08
19.What is 0.6 × 0.4?
Medium
- A2.4
- B0.24
- C0.024
- D24
20.Which of these is equal to 3/5?
Medium
- A0.35
- B0.6
- C0.53
- D0.65
21.Which of these is equal to 0.45?
Medium
- A9/20
- B45/100
- C4/5
- D5/9
22.Which of these is equal to 0.07?
Medium
- A0.7%
- B7%
- C70%
- D0.07%
23.Which of these is equal to 25%?
Medium
- A0.025
- B0.25
- C2.5
- D25
24.Which of these is equal to 2/5?
Medium
- A20%
- B40%
- C50%
- D25%
25.Which is the correct decimal expansion of 1/8?
Hard
- A0.125
- B0.18
- C0.25
- D0.128
26.Which of these fractions is equivalent to the recurring decimal 0.666...?
Medium
- A2/3
- B3/5
- C6/10
- D5/8
27.A plank of wood is 2.4 m long. It is cut into pieces each 0.15 m long. How many pieces are produced?
Medium
- A12
- B16
- C18
- D24
28.A rectangle has length 3.2 cm and width 1.5 cm. What is its area?
Medium
- A4.8 cm²
- B3.6 cm²
- C4.7 cm²
- D6.4 cm²
29.A shirt is priced at £24. In a sale, the price is reduced by 15%. What is the sale price?
Hard
- A£3.60
- B£20.40
- C£21.60
- D£18.00
30.Which of the following calculations give a result greater than 1? (select all that apply)
Medium
- A0.6 × 2
- B0.75 + 0.2
- C1.2 ÷ 0.5
- D0.9 × 0.9
- E0.5 + 0.4
31.Which of the following are equivalent to 0.125? (select all that apply)
Hard
- A1/8
- B12.5%
- C1/4
- D125/1000
- E0.1250
32.When you divide a decimal by 100, the digits move two places to the left.
Medium
True or false?
33.The decimal 0.121212... is a recurring decimal.
Medium
True or false?
34.What is the base (radix) of the decimal system?
Easy
- A10
- B2
- C8
- D16
35.Which of the following are alternative names for the decimal system? (select all that apply)
Medium
- Abase-ten
- Bdenary
- Cdecenary
- Dbinary
- Ehexadecimal
36.The decimal digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
Easy
True or false?
37.Which of the following is a non-positional base-ten numeral system?
Easy
- ARoman numerals
- BHindu–Arabic numerals
- CBinary numerals
- DHexadecimal numerals
38.In the context of decimal notation, what does the word 'decimal' often refer to in English?
Medium
- AThe digits after the decimal separator
- BThe digits before the decimal separator
- CThe decimal separator itself
- DThe minus sign
39.Which of the following is a decimal fraction?
Medium
- A\frac{3}{10²}
- B\frac{1}{3}
- C\frac{2}{7}
- D\frac{5}{6}
40.A decimal fraction is a fraction of the form \frac{a}{10ⁿ}, where a is an integer and n is a non-negative integer.
Easy
True or false?
41.Complete the table converting between fractions, decimals and percentages. Write each answer in its simplest form.
Medium
Fraction: 1/4, Decimal: ____, Percentage: ____%
42.What is the name for a decimal that has a finite number of digits after the decimal point?
Easy
43.Fill in the blank: the decimal 4.690 has ___ decimal places.
Easy
The decimal 4.690 has ____ decimal places.
44.Write 0.75 as a fraction in its simplest form.
Medium
45.Calculate 0.5 × 0.3.
Medium
46.Work out 4.6 × 1000.
Easy
47.Work out 0.087 × 100.
Easy
48.Work out 5.3 ÷ 100.
Easy
49.Work out 1.25 + 0.375.
Medium
50.Work out 4.2 − 1.75.
Medium
51.Work out 0.07 × 0.9.
Medium
52.Work out 3.6 ÷ 0.4.
Medium
53.Explain how to convert a recurring decimal such as 0.777... into a fraction.
Hard
54.Explain how to convert the terminating decimal 0.6 into a fraction in its simplest form.
Hard
55.Explain why 0.999... is equal to 1.
Medium
56.Explain how to convert the recurring decimal 0.555... into a fraction in its simplest form.
Hard
57.Explain how to convert the terminating decimal 0.45 into a fraction in its simplest form.
Hard
58.Write 0.75 as a fraction in its simplest form.
Medium
- A3 /4
- B2 /4
- C5 /4
- D4 /4
59.Calculate 0.5 × 0.3.
Medium
- A0.15
- B1.15
- C-0.85
- D2.15
60.Work out 4.6 × 1000.
Easy
- A5520
- B4140
- C5060
- D4600
61.Work out 0.087 × 100.
Easy
- A9.7
- B10.7
- C8.7
- D7.7
62.Work out 5.3 ÷ 100.
Easy
- A1.053
- B-0.947
- C2.053
- D0.053
63.Work out 1.25 + 0.375.
Medium
- A1.625
- B0.625
- C2.625
- D3.625
64.Work out 4.2 − 1.75.
Medium
- A2.45
- B1.45
- C4.45
- D3.45
65.Work out 0.07 × 0.9.
Medium
- A1.063
- B2.063
- C0.063
- D-0.937
66.Work out 3.6 ÷ 0.4.
Medium
- A10
- B8
- C11
- D9

Notes de révision

## Decimals and Converting Fractions, Decimals and Percentages ### Place value in decimals A decimal numeral uses a decimal point to separate whole-number digits from fractional digits. In 20.70828, the digits after the point are tenths, hundredths, thousandths, and so on. The value of each digit depends on its position. - 0.1 = one tenth = \(\frac{1}{10}\) - 0.01 = one hundredth = \(\frac{1}{100}\) - 0.001 = one thousandth = \(\frac{1}{1000}\) Trailing zeros after the decimal point do not change the value: 5.2 = 5.20 = 5.200. Leading zeros before the number do not change it either: 3.14 = 03.14. ### Multiplying and dividing by powers of 10 Multiplying by 10, 100, 1000, ... moves the digits left; dividing moves them right. - 3.45 × 10 = 34.5 - 3.45 × 100 = 345 - 3.45 ÷ 10 = 0.345 - 3.45 ÷ 100 = 0.0345 ### Calculating with decimals Line up the decimal points when adding or subtracting. For multiplication, multiply as whole numbers, then count the total decimal places in the question and put that many in the answer. For division, make the divisor a whole number by multiplying both numbers by the same power of 10. ### Converting between fractions, decimals and percentages - Fraction to decimal: divide the numerator by the denominator. Example: \(\frac{3}{4} = 3 \div 4 = 0.75\). - Decimal to percentage: multiply by 100. Example: 0.75 × 100 = 75%. - Percentage to decimal: divide by 100. Example: 75% ÷ 100 = 0.75. - Decimal to fraction: use place value. Example: 0.75 = \(\frac{75}{100} = \frac{3}{4}\). - Percentage to fraction: write over 100 and simplify. Example: 75% = \(\frac{75}{100} = \frac{3}{4}\). ### Terminating and recurring decimals A terminating decimal stops after a finite number of digits, e.g. 0.25. A recurring decimal repeats a block of digits forever. Dot notation shows the repeating part: - 0.333... = 0.\(\dot{3}\) - 0.121212... = 0.\(\dot{1}\dot{2}\) - 5.123144144144... = 5.123\(\dot{1}4\dot{4}\) A decimal represents a rational number (a fraction of two integers) if and only if it terminates or recurs. ### Ordering a mixed list Convert everything to the same form (usually decimals) before comparing. Example: order 0.6, \(\frac{2}{3}\), 65%, \(\frac{3}{5}\). - 0.6 = 0.6 - \(\frac{2}{3}\) = 0.666... - 65% = 0.65 - \(\frac{3}{5}\) = 0.6 So the order is 0.6 = \(\frac{3}{5}\), 65%, \(\frac{2}{3}\). ### Key facts - Decimal system = base-ten system. - Decimal fractions are fractions of the form \(\frac{a}{10^n}\), where a is an integer and n is a non-negative integer. - More decimal places can mean greater precision: 1.32 mg suggests a range 1.315 to 1.325 mg, while 1.320 mg suggests 1.3195 to 1.3205 mg.

Decimal system: A numeral system that uses ten as its base (also called base-ten, denary or decenary).
Decimal point: The dot (or comma) that separates the whole-number part from the fractional part of a decimal numeral.
Place value: The value of a digit based on its position in a number; after the decimal point, digits represent tenths, hundredths, thousandths, etc.
Decimal fraction: A fraction of the form \(\frac{a}{10^n}\), where a is an integer and n is a non-negative integer; these can be written exactly as finite decimals.
Terminating decimal: A decimal expansion that reaches a point where all remaining digits are zero, so it can be written with a finite number of digits.
Recurring (repeating) decimal: An infinite decimal that, after some place, repeats the same sequence of digits forever, e.g. 0.333... or 5.123144144144...
Dot notation: A way of writing recurring decimals by placing dots over the first and last digits of the repeating block, e.g. 0.\(\dot{3}\) or 0.\(\dot{1}\dot{2}\).
Rational number: A number that can be written as the quotient of two integers. A decimal is rational if and only if it terminates or recurs.
Percentage: A fraction expressed out of 100; to convert a decimal to a percentage, multiply by 100.

Cartes mémo

What is a decimal system?

A decimal system is a numeral system that uses ten as its base (radix). It is also called base-ten, denary or decenary.

What is decimal notation?

Decimal notation is the way of denoting numbers in a decimal system, using ten digits and a decimal separator.

What is a decimal numeral?

A decimal numeral is the notation of a number in a decimal numeral system. It may include a decimal separator to show non-integer parts.

What is a decimal fraction?

A decimal fraction is a fraction of the form a/10ⁿ, where a is an integer and n is a non-negative integer. These are exactly the numbers that can be represented by a finite decimal.

What is a terminating decimal?

A terminating decimal is a decimal expansion that reaches a point where all remaining digits are zero, so the expansion can be stopped. For example, 0.25 is terminating.

What is a repeating decimal?

A repeating decimal is an infinite decimal that, after some place, repeats the same sequence of digits indefinitely. For example, 5.123144144144144... = 5.123144 with dots over the repeating part.

How do you write a repeating decimal using dot notation?

Place a dot over the first and last digits of the repeating block. For example, 0.333... is written as 0.̇3̇ (dot over the 3), and 0.1666... as 0.1̇6̇ (dot over the 1 and 6).

When does an infinite decimal represent a rational number?

An infinite decimal represents a rational number (a quotient of two integers) if and only if it is a repeating decimal or has a finite number of non-zero digits.

What is the decimal separator in English-speaking countries?

In English-speaking countries, the decimal separator is a dot (.). In many other countries, a comma (,) is used.

What are the decimal digits in Arabic numerals?

The decimal digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

How do you multiply a decimal by 10?

To multiply by 10, move the decimal point one place to the right. For example, 3.14 × 10 = 31.4.

How do you divide a decimal by 10?

To divide by 10, move the decimal point one place to the left. For example, 3.14 ÷ 10 = 0.314.

How do you multiply a decimal by 100?

To multiply by 100, move the decimal point two places to the right. For example, 0.25 × 100 = 25.

How do you divide a decimal by 100?

To divide by 100, move the decimal point two places to the left. For example, 25 ÷ 100 = 0.25.

What is the place value of the digit 7 in 3.147?

The digit 7 is in the thousandths place, so its value is 7/1000 or 0.007.

How do you convert a fraction to a decimal?

Divide the numerator by the denominator. For example, 3/4 = 3 ÷ 4 = 0.75.

How do you convert a decimal to a fraction?

Write the decimal as a fraction with denominator 10, 100, 1000, etc., depending on the number of decimal places, then simplify. For example, 0.6 = 6/10 = 3/5.

How do you convert a decimal to a percentage?

Multiply the decimal by 100 and add the % sign. For example, 0.75 × 100 = 75%.

How do you convert a percentage to a decimal?

Divide the percentage by 100. For example, 75% ÷ 100 = 0.75.

How do you convert a fraction to a percentage?

First convert the fraction to a decimal by dividing, then multiply by 100 and add the % sign. For example, 3/4 = 0.75 = 75%.

How do you convert a percentage to a fraction?

Write the percentage as a fraction with denominator 100, then simplify. For example, 75% = 75/100 = 3/4.

What is 1/2 as a decimal and a percentage?

1/2 = 0.5 = 50%.

What is 1/4 as a decimal and a percentage?

1/4 = 0.25 = 25%.

What is 3/4 as a decimal and a percentage?

3/4 = 0.75 = 75%.

What is 1/5 as a decimal and a percentage?

1/5 = 0.2 = 20%.

What is 1/8 as a decimal and a percentage?

1/8 = 0.125 = 12.5%.

What is 1/3 as a decimal and a percentage?

1/3 = 0.333... (repeating) = 33.3...% (repeating).

What is 2/3 as a decimal and a percentage?

2/3 = 0.666... (repeating) = 66.6...% (repeating).

How do you order a mixed list of fractions, decimals and percentages?

Convert them all to the same form (e.g., decimals), then compare and order. For example, to order 0.6, 2/3, 65%, convert to decimals: 0.6, 0.666..., 0.65, so the order is 0.6, 65%, 2/3.

What does the number of decimal places indicate about precision?

The number of decimal places given indicates the precision to which a quantity is known. For example, 1.32 mg suggests the true mass is between 1.315 mg and 1.325 mg, while 1.320 mg suggests it is between 1.3195 mg and 1.3205 mg.

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