DECIMAL NUMBERS 1

To express quantities in between two integers, we use decimal numbers thousandshundredstens units tenthshundredthsthousandthstenthousandthshundredthousandthsmillionths

The decimal part represents an amount smaller than the unit. The decimal place values have the same structure as that of the integer part.

A unit of any place value is divided into ten units of the place value immediately below.

Types of decimal numbers

Twenty-eight units and three thousand, seven hundred and fifty-eight ten thousandths

It is important to be able to differentiate between different types of decimal numbers, as you will find them in measurements, results of operations and problems.

• Terminating decimals have a limited number of decimal places. 4.75 two decimal places

• Recurring decimals have an infinite number of decimal places which are repeated periodically. There are two types:

Pure recurring decimals: Mixed recurring decimals: 7.151515… = 7. 15 # 8.24666… = 8.24

• Non-terminating and non-recurring decimals have an infinite number of decimal places which do not repeat periodically.

2 = 1.4124135…

Practical rule

To compare two decimal numbers, we compare the corresponding decimal places, one by one, starting from the left.

Representing and ordering decimal numbers

Each decimal number is represented with a point on the number line. Each point on the line can be located by a decimal number.

–1.263 < –0.4 < 0.6751 < 1.55 < 2.753

Decimal numbers are ordered on the number line. If we choose any two numbers, the smaller of the two is on the left and the bigger is on the right.

1 Write the quantities in the table as words:

2 Write the following quantities as words: a) Three units and five hundredths. b) Forty-three thousandths. c) Eight thousandths. d) Two hundred and nineteen millionths. e) Twenty-three millionths. f) Fourteen ten millionths.

6 Look at the number line below. The red dot represents life expectancy in Spain in the year 2019. The yellow dot represents life expectancy in 2020.

3 Write in figures.

4 Look at the following decimal numbers: 1.292929… 3 = 1.7320508… a) Which ones are terminating decimals? b) Which ones are pure recurring decimals? c) Which ones are mixed recurring decimals? d) Which ones are neither terminating nor recurring?

82 years 83 years 84 years a) Write this data to two decimal places. b) Did it improve or worsen from 2019 to 2020? What do you think the cause was? a) 7.4; 6.9; 7.09; 7.11; 5.88 b) 3.9; 4.04; 3.941; 3.906; 4.001 c) 0.039; 0.01; 0.06; 0.009; 0.075 d) 11.99; 11.909; 11.009; 12.01; 11.91

7 Put these numbers in order, from lowest to highest.

8 Copy and complete the boxes in your notebook with the signs <, > or =, as appropriate. a)

Focus on English

meaningful: useful, important.

Approximating a decimal number to a certain decimal place

Calculations sometimes result in numbers with too many decimals. These are difficult to work with and provide little meaningful* information. In these cases, we take a smaller number of decimal places. This gives an approximate value, which is easier to work with.

Practise rounding decimal numbers.

When rounding off, if the first deleted digit is 5, we round up by adding one to the digit before it. Rounding

The result 1.8333… is closer to 1.83 than to 1.84.

Rounding off: Each jar of jam costs €1.83.

The result 1.8375 is closer to 1.84 than to 1.83.

Rounding off: Each packet of coffee costs €1.84

Rounding off involves deleting decimal figures from a certain decimal place value, and adding one unit to the last digit when the first one we delete is 5 or greater than 5.

Error bound

When we round off we give an approximate value, so we are making an error on purpose.

The error made in rounding off is less than half a unit of the place value we have rounded to.

In both cases, we have rounded off to the nearest hundredths, and the error made is less than five thousandths; in other words, less than half a hundredth.

We say that the error bound is half a hundredth.

Practical rule

To find decimal numbers between two given decimals, we increase the number of decimal places by adding zeros to the right of the last digit.

5.09 < … < 5.1

5.090 < 5.095 < 5.100

Numbers

• Let’s take any two decimals, for example 5.5 and 5.9. Obviously, there are other numbers between the two.

• Now we take two consecutive decimals from between the previous ones, for example, 5.5 and 5.6.

The difference between the two numbers is one tenth, which is divided into ten hundredths.

This reasoning can continue indefinitely and repeat for any other pair of numbers. Between two decimal numbers there are an infinite number of other decimals.

Consolidating ideas

1 In your notebook, round off the number 2.83516: a) To the nearest unit. 8 b) To the nearest tenths. 8 c) To the nearest hundredths. 8 … d) To the nearest thousandths. 8 … a) Find three numbers between 2.58 and 2.59.

2 Copy and complete.

2.580 < … < … < … < 2.590 b) Find three numbers between 3.4 and 3.41.

3. … < … < … < … < 3. … c) Find three numbers between 0.59 and 0.6. 0.59… < … < … < … < 0.6…

Let's practise!

9 Round off the number . 682 # : a) To the nearest unit. b) To the nearest tenths. c) To the nearest d) To the nearest hundredths. thousandths. a) 5.48 b) 2.8346 c) 3.057

10 Round off to the nearest tenths.

2 Find a number between 4.09 and 4.1: 4.09 4.1 a) 6.284 b) 1.53369 c) 0.79462

11 Round off to the nearest hundredths.

12 Find a decimal number between: a) 2.2 and 2.3 c) 1.59 and 1.6 b) 4.01 and 4.02 d) 8 and 8.1

Operating With Decimal Numbers 2

You already know how to add, subtract and multiply with decimals. Let’s review the operations in this water bill, as well as the rounding off of the results.

anayaeducacion.es

Practise calculating with decimal numbers in your head.

The calculations are shown in the margin and are summarised in these expressions: anayaeducacion.es Practise dividing with decimal numbers.

• To add or subtract decimal numbers, we put them in a column matching the corresponding decimal places.

• To multiply decimal numbers, we multiply them as if they were integers. We then separate as many decimal places in the product as the sum of the number of decimal places of the two factors.

Dividing decimal numbers

We will now review the different types of divisions with decimal numbers. For each one, we will start with a problem that will make the operation more meaningful.

➜ divisions with an integer divisor

Problem 1

A 70-litre can of oil is filled using 16 bottles, which are all the same size. How much oil is in each bottle?

Solution: Each bottle contains 4.375 litres of oil.

Problem 2

Remember

When rounding off, if the first deleted digit is 5 or greater, we round up by adding one to the digit before it. 1.19666… rounding off

A bag of oranges weighing 3 kilos costs €3.59. What is the price per kilo?

One kilo costs 1196 ! 8 Rounding off: €1.20

To get decimal places in the quotient:

• When we lower the tenths digit of the dividend, we put the decimal point in the quotient and then continue the division.

• If we do not have enough decimal places in the dividend, we add zeros to get to the desired approximation.

➜ divisions with a decimal divisor

Problem 3

Remember

If we multiply the dividend and the divisor by the same number, the quotient does not change. 6 :

Target 5.2. We pay €6 for 2.5 m of fabric to make a banner demanding gender equality.

How much does a metre of the fabric cost?

Solution: One meter of the fabric costs €2.40.

Problem 4

A 1.25 m2 sheet of chipboard weighs 8 kilos and 100 grams.

How much does one square metre weigh?

Solution: A square metre weighs 6.48 kg.

When there are decimals in the divisor:

• We multiply both the dividend and the divisor by the unit followed by as many zeros as there are decimal places in the divisor.

• The new division has the same quotient and an integer divisor.

Combined operations

In expressions involving decimal numbers with brackets and combined operations, we follow the same rules as we do with integers with regard to the priority of operations, the rule of signs, etc.

Priority of operations:

• First, brackets.

• Then, multiplications and divisions.

• Finally, additions and subtractions.

Consolidating ideas

1 Copy and complete to get an equivalent division, but without decimals in the divisor. Then, do the operations.

Help

1 Example:

2 Look at the result a calculator gives when we divide 2.5 : 6. Then, copy and complete the problems in your notebook, rounding off the answers as appropriate.

2.5 / 6 = 8 {∫≠Ÿ¢‘\\\\\} a) 2.5 kg of silver is used to make six trophies.

Each trophy contains … kilos of silver. 8 Rounding off: … grams b) 2.5 kg of potatoes are used to make six tortillas

Each tortilla contains … kilos of potatoes. 8 Rounding off: … grams

3 Look at the combined operation, copy and complete. 3 – (1.5 + 1.54) : (4.23 – 2.33)

2 Example:

• 4.75 kg of flour makes 16 cakes.

• 4.75 kg of gold makes 16 bars. We solve using a calculator: 4.75 / 16 = 8 {∫≠Ÿ“£\°|∞‚}

• Each cake contains 0.3 kg of flour.

• Each bar contains 0.297 kg of gold.

3 Example: (3 – 1.9) · (1.43 + 1.07) – 1.15 1.1 · 2.5 – 1.15 2.75 – 1.15 1.6 a) The total in section A for a consumption of 40 m3 would be: 10 · 0.03 + … · 0.65 + … · 1.93 = …. + … + … = … b) Calculate the ‘total due’ from the bill above for a family with a consumption of 40 m3 that does not get any discount. c) Does all the water consumed cost the same? Why do you think this is?

4 Look at the bill from page 38 again. Copy and complete.

4 A water consumption of 55 m3 is divided as follows:

Up to 10 m3 8 10 m3

From 11 to 30 m3 8 19 m3

From 31 to 60 m3 8 26 m3

Total 8 55m3 decimal numbers. a) 0.75 + 0.25 b) 0.75 – 0.25 c) 1.80 + 1.20 d) 1.80 – 1.20 e) 2.30 + 1.80 f ) 2.30 – 1.80 g) 3.50 + 1.75 h) 3.50 – 1.75 a) 2.37 + 0.356 b) 5.86 – 1.749 c) 13.2 + 4.08 + 2.635 d) 15.4 – 6.843 e) 7.04 + 12.283 + 0.05 f ) 0.35 – 0.0648 a) 2.6 · 100 b) 5.4 : 10 c) 0.83 · 10 d) 12 : 100 e) 0.0048 · 1 000 f ) 350 : 1 000 a) 6.3 · 1.24 b) 0.44 · 2.375 c) 0.016 · 0.0025 d) 143 · 0.068 e) 5.48 · 2.63 f ) 0.15 · 1.01 a) 5.2 : 0.8 = 52 : … b) 3 : 0.004 = … : 4 c) 6.31 : 2.5 = … : 25 d) 2.4 : 1.638 = 2 400 : … e) 0.005 : 0.02 = 5 : … f) 0.12 : 0.0012 = 1 200 : … a) 8 : 6 b) 218 : 16 c) 3 : 4 d) 12 : 536 e) 149.04 : 23 f ) 2.58 : 15 a) 6 : 0.2 b) 13 : 0.75 c) 53 : 4.11 d) 4 : 0.009 e) 45.6 : 3.8 f ) 23.587 : 5.1 g) 2.549 : 8.5 h) 6.23 : 0.011 a) 5 : 6 b) 7 : 9 c) 6 : 3.5 d) 2.7 : 5.9 a) 2.37 – 1.26 + 0.8 – 0.35 b) 2.50 – 1.25 – 1.75 – 0.20 c) 13.48 – 10.7 + 5.328 – 6.726 d) 5.6 – 8.42 – 4.725 + 1.48 a) 6.2 – (7.2 – 4.63) b) (12.85 – 7.9) – (6.2 + 3.28) c) 5.6 – [4.23 – (5.2 + 1.75)] a) 3.6 – 1.2 · 0.6 – 4.5 : 1.8 b) 3.6 – 0.5 · (4 – 2.26) c) 0.75 : (2.65 – 1.15) – 1.1 d) (0.5 + 0.1) · (0.5 – 0.1) – (0.6 – 0.4) · (0.6 + 0.4) e) 5.4 – 1.5 · [3.2 + 10 · (0.63 – 1.25)] a) Multiplying by 0.1 is the same as dividing by... b) Dividing by 0.1 is the same as multiplying by... c) Multiplying by 0.5 is the same as... d) Dividing by 0.5 is the same as... e) Multiplying by 0.25 is the same as... f) Dividing by 0.25 is the same as... a) 25.197 · 9.86 b) 142.36 · 0.49 c) 181.046 : 6.16 d) 33.44 : 0.511 a) A packet of 500 sheets of paper weighs 652 g. What does one sheet of paper weigh? b) A chicken costs €3.49/kg. How much will a chicken that weighs one kilo and 850 grams cost? c) A 2-metre-long piece of wood is cut into seven equal pieces. What is the length of each piece? d) A car consumes 50 litres of petrol over 837 km. How much does it consume per 100 kilometres?

Let's practise!

1 Calculate mentally.

2 Calculate.

3 Remind yourself how to multiply and divide by the unit followed by zeros and complete.

4 Calculate.

5 Copy and complete so that each equality is true.

6 Calculate the exact quotient or to a maximum of three decimal places.

7 Replace each division with one that is equivalent, but has an integer divisor. Then, calculate the exact quotient or to a maximum of three decimal places.

8 Round off each quotient to the nearest hundredths.

9 Solve.

10 Calculate.

11 Do the operations and solve.

12 Experiment, think of some examples, and then copy and complete in your notebook.

13 Calculate mentally.

14 Estimate mentally, without using decimals, and then check with a calculator.

You should make an error of 2 units or less.

15 Solve with a calculator and round off to the place value that you consider appropriate.