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📐 Mathematics - Class 7
Lesson 16 · Fractions and Decimals

Decimals and Simplification

MathematicsSubject
13 minEstimated read
A decimal is a fraction with the bottom already chosen

A decimal is a fraction whose bottom is always a power of ten, agreed in advance so nobody has to write it.

Place value after the point

After the point, the same rule keeps going: tenths, hundredths, thousandths. The pattern has just been carried on past the units column.

Place Worth In 3.482
Units13 units
Tenths1/104 tenths
Hundredths1/1008 hundredths
Thousandths1/10002 thousandths
More digits does not mean bigger

Is 0.7 or 0.65 larger? 0.7 is seven tenths while 0.65 is six tenths and a bit. Writing 0.7 as 0.70 makes it obvious.

Adding and subtracting

Line up the decimal POINTS, not the right hand ends, and fill short columns with zeros.

Why multiplying moves the point

Tenths times tenths gives hundredths, exactly as multiplying the bottoms of two fractions gives a bigger bottom.

0.3 times 0.4 is not 1.2

Both numbers are less than 1, so the answer must be less than either. 0.12 is right while 1.2 is ten times too big.

Dividing by a decimal

Shift the point in both numbers by the same amount: 4.8 ÷ 0.6 becomes 48 ÷ 6, which is 8.

Rounding is a decision, not a mistake

Look at the next digit only. Round at the END of a calculation, never in the middle.

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Worked Example 1

A shopper buys goods costing Rs 245.50, Rs 8.75 and Rs 120.

Write them with the points lined up245.50, 8.75, 120.00 with all three points in one column
Add column by columnRs 374.25
Why the zeros matterWithout them, someone might leave 120 hanging in the wrong columns entirely.
Worked Example 2

A quantity of 0.4 is needed at a rate of 0.3 per unit.

Multiply as if there were no points3 × 4 = 12
Count the digits after the pointsOne plus one is two, so the answer is 0.12
Sense check0.12 passes and 1.2 fails the check.
Where the rule comes from3/10 × 4/10 = 12/100. The counting rule is just a shortcut for that.
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Activity 16.2

The shop receipt check

What you needA real shop receipt, or a list of five prices, and a pencil. No calculator until the end.
MethodCopy the prices into a column with the points exactly under one another, writing whole amounts with .00.
Then estimateRound every price to the nearest rupee, add those in your head, and compare with your exact answer.
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1

A decimal is a fraction whose bottom is always a power of ten.

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1Why is a decimal just a fraction, and what difference does that make?
2Why is 0.7 larger than 0.65, when 65 is larger than 7?
3Why must you line up the decimal points rather than the right hand ends when adding?
4Explain the rule for multiplying decimals and why it works.
5Why does dividing 4.8 by 0.6 give an answer bigger than 4.8?

Question 1 of 10

1In 3.482, the 8 is worth
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Slide 1 of 4
Unit 16 · Decimals

Decimals

Mathematics · Grade 7

What You Will Be Able To Do

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Name the value of every digit after the decimal point

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Compare decimals correctly, even when one has more digits

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Add, subtract, multiply and divide them without losing the point

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See a decimal as a fraction whose bottom is already chosen

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A decimal is not a new kind of number. It is a fraction whose bottom is always a power of ten, agreed in advance so nobody has to write it.

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Presenter notes: Open with a question that has an obvious wrong answer. Write 0.7 and 0.65 on the board and ask which is larger. A good number of the class will say 0.65 because it has more digits, and that is worth surfacing now rather than discovering in an exam. Then ask why money and measurements are always written as decimals rather than fractions. The answer to both questions is the same idea, and it takes about a minute to explain once the lesson gets going.