Maths
Decimals
Decimals play an important part in everyday life - particularly through the use of money - so it's crucial that you know how to use them.
This Revision Bite covers:
We use a decimal point to separate units from parts of a whole (like tenths, hundredths, thousandths, etc).
In 52.13, the value of the figure 1 is 1/10 , and the value of the figure 3 is 3/100.
When ordering numbers, always compare the left digits first.
Eg Which is greater, 2.301 or 2.32?
Units | Tenths | Hundredths | Thousandths | |
---|---|---|---|---|
2 | . | 3 | 0 | 1 |
2 | . | 3 | 2 |
Both numbers have two units and three tenths, but 2.301 has no hundredths, whereas 2.32 has two hundredths. Therefore, 2.32 is greater than 2.301.
Another way to look at it is to add a zero to the end of 2.32 (this doesn't change its value as it's after the decimal point).
The two numbers are now 2.320 and 2.301 and it is quite easy to see that 2.320 is bigger (just as 2 320 is bigger than 2 301).
Q1. In the number 3.546, what is the value of the figure 4?
Q2. Place the following numbers in order, smallest first: 3.2, 3.197, 3.02, 3.19
A1. The value of the figure 4 is 4/100.
A2. Did you get 3.02, 3.19, 3.197, 3.2?
All the numbers have three units, so start by comparing the tenths. 3.02 has no tenths, 3.197 and 3.19 have one tenth, and 3.2 has two tenths. Therefore, 3.02 is the smallest and 3.2 is the largest.
You could also write the numbers as 3.020, 3.190, 3.197 and 3.200 and compare them.
When adding and subtracting decimals, remember is to keep the decimal points in line in the question and the answer.
David is doing some DIY. He buys a 2m length of wood. He needs to cut two pieces of wood - one of length 0.6m and one of length 1.02m.
What is the total length of wood that David needs to cut?
Add 0.6 and 1.02.
So the total length of wood that David needs to cut is 1.62m.
You can add zeros to the end of a decimal without affecting its value, so 0.6 is the same as 0.60.
David originally had 2m of wood. What is the length of the piece of wood that is left?
David cut off 1.62m, so we need to calculate 2 - 1.62
So there is 0.38m of wood left.
When a decimal is multiplied by 10, every figure moves one place to the left.
When multiplying by 100, every figure moves two places to the left.
When multiplying by 1000, every figure moves three places to the left.
Which is bigger: 0.005 × 10 or 0.0004 × 1000?
The answer is 0.0004 × 1000
If you didn't get it right, remember that multiplying by 10 moves each digit one place to the left.
Therefore, 0.005 × 10 = 0.05.
Multiplying by 1000 moves each digit three places to the left.
Therefore, 0.0004 × 1000 = 0.4.
0.4 is bigger than 0.05.
When you divide by 10, every figure moves one place to the right. Hundreds become tens, tens become units, units become tenths and tenths become hundredths.
When you divide by 100, every figure moves two places to the right.
When you divide by 1000, every figure moves three places to the right.
Multiplying a decimal by a whole number is the same as multiplying two whole numbers. Remember:
Calculate:
a) 2.43 × 7
b) 2.4 × 5
a) There were two digits after the decimal point in the question (4 and 3), so you must have two digits after the decimal point in the answer.
b) There was one digit after the decimal point in the question, so you must have one digit after the decimal point in the answer. The answer is therefore 12.0, but this can then be given as 12.
Check that you have a sensible answer by finding an approximate solution.
In the above example you were asked to calculate 2.4 × 5.
2 × 5 = 10, so you are looking for an answer which is slightly bigger than 10. So an answer of 12 seems sensible.
Remember to keep the decimal points aligned in the question and the answer.
Work out 4.05 divided by 9
Solution:
Work out 2.4 divided by 5
Solution:
It is sometimes necessary to add a '0' or '0's to the end of a decimal, as in this example (2.40 is the same as 2.4 but the question stays the same)
The multiplication sign can be replaced by 'lots of'.
For example,
2 × 3 means 2 lots of 3
6 × 8 means 6 lots of 8
So, ^{1}/_{2} × 10 means ^{1}/_{2} of 10
And ^{1}/_{3} × 12 means ^{1}/_{3} of 12
When you multiply by a number greater than 1, you get an answer that is greater than the original number. But when you multiply by a number between 0 and 1, the answer is smaller than the original number.
In general:
m × 1/n = m ÷ n
8 × ^{1}/_{4} = 8 ÷ 4 = 2
20 × ^{1}/_{5} = 20 ÷ 5 = 4
Imagine that you had 10 bars of chocolate that you wanted to share amongst some children.
If you gave the children 2 bars each, you would have enough for 5 children.
10 ÷ 2 = 5
If you gave the children ^{1}/_{2} bar each, you would have enough for 20 children.
10 ÷ ^{1}/_{2} = 20
Can you see what's happening?
10 ÷ 2 = 5
10 ÷ ^{1}/_{2} = 20
When you divide by a whole number the answer is less than the original number. When you divide by ^{1}/_{2} the answer (20) is greater than the original number (10).
It's the opposite of multiplying. When we divide by a number greater than 1, we get an answer that is less than the original number. But when we divide by a number between 0 and 1 the answer is larger than the original number.
So, 10 ÷ ^{1}/_{2} = 20
Similarly, 10 ÷ ^{1}/_{3} = 30 and 10 ÷ ^{1}/_{4} = 40
In general:
m ÷ ^{1}/_{n} = mn
Q1. What is 10 ÷ ^{1}/_{7} ?
Q2. Find the value of: 4 ÷ ^{1}/_{3}
A1. 70
A2. 4 ÷ ^{1}/_{3} = 4 × 3 = 12
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