### Key Concepts

■ Addition of decimals

■ Subtraction of decimals

■ Multiplication of decimals

**Introduction**:

## 1.1 Addition of decimals

### Decimal number:

A number with a whole part and a fractional part separated by a decimal point is called the decimal number. Decimal point is denoted by a symbol dot (.).

#### Like decimal numbers:

The numbers that have same number of digits on the right side of the decimal point are called like decimals.

Example: 7.05, 12.45, 125.16, etc., are like decimals, because all the numbers have 2 places of decimal numbers.

#### Unlike decimal numbers:

The numbers that do not have same number of digits on the right side of the decimal point are called unlike decimals.

**Example:** 5.4, 32.74, 54.025, etc., are unlike decimals, because the numbers have different place values after the decimal point.

**Note:** We can convert unlike decimals into like decimals by adding required number of zeros after the decimal point.

**How to add decimal numbers?**

We can add the decimal numbers with the following steps:

Step 1: First, check the decimals, whether they are like or unlike. Convert the unlike decimals into like decimals by adding required number of zeros after the decimal point.

Step 2: Decimals can be written one below the other according to their places by lining up the decimal point.

Step 3: Add the digits from right to left.

Step 4: Put the decimal point in the final answer.

**Example 1:** Add 3.25 and 2.14.

**Solution:**

Given decimal numbers are like decimals, we can add these two decimal numbers as regular addition.

∴∴

The sum of the two decimal numbers is 5.39.

**Example 2:** Add 7.4 and 3.24

**Solution:**

We can observe that the given decimals are unlike, we have to convert them to like decimals by adding zeros on the right side of the decimal point.

3.24 has two digits after the decimal point, whereas 7.4 has only one digit.

If we add one zero to 7.4, then it becomes 7.40. Now, both the numbers have equal number of digits to perform the addition.

∴∴

The sum of the two decimal numbers is 10.64.

**How to add decimals with whole numbers?**

Let us consider the following decimal number with whole numbers to add.

**Example 3:** Add 6 and 7.24.

**Solution:**

Here, 6 is a whole number and 7.24 is a decimal number.

Place a decimal point after 6 and add two zeros, so that the whole number 6 becomes a decimal number.

6 = 6.00

Now, add the two decimal numbers to find the sum.

∴∴

The required sum of the two decimal numbers is 13.24.

**How to add decimals by regrouping?**

#### Regrouping:

The sum of the addends that is greater than 9, then regroup it by carrying over the extra digit to its preceding column.

**Example 4:** Add 15.62 and 13.65.

**Solution:**

Adding the decimals by regrouping.

**How to add with estimation?**

**Example 5:**

Jennifer and Anna complete the running race of 19.7 meters. Anna finishes the race in 0.36 seconds longer than Jennifer. Find Anna’s time in the race.

**Solution:**

Find the sum of 19.7 and 0.36.

Estimate each addend by rounding to its place values.

19.7 rounds to 20

0.36 rounds to 0.4

Sum = 20 + 0.4 = 20.4

Find the sum by lining up the place values.

Add a zero to the decimal number 19.8, then it becomes 19.80 which is a like decimal.

Now, regroup and add both the like decimal numbers.

Anna finishes the race in 20.16 seconds. The sum 20.16 is close to the estimate 20.4.

**1.2 ****Subtraction of decimals**

Subtraction of decimal numbers is also same as addition of decimal numbers.

**Example 1:**

Peter swam a swimming race in 20.7 seconds. Mark took 0.258 seconds to complete the race before Peter. What was Mark’s time in the race?

**Solution:**

Find the difference of 20.7 – 0.258.

Estimate the difference by rounding.

0.258 is close to 0.3.

20.7 – 0.3 = 20.4

To find the difference of 20.7 and 0.258, we have to line up the place values.

We have to convert the unlike decimal 20.7 into like decimal. So, adding two zeros after decimal point and next to 7.

20.7 = 20.700, now regroup and subtract the like decimals.

Mark swam the race in 20.442 seconds.

**1.3 ****Multiplication of decimals**

### Multiplication of decimals with whole numbers:

The following steps has to be followed when we multiply decimals with whole numbers:

**Step 1:** First, multiply the two numbers before the decimal point.

**Step 2:** Calculate the number of decimal places in the product after multiplying the two numbers.

**Step 3:** Place the decimal point in the product.

**Example 1:** Multiply 13 and 7.5.

**Solution:**

### Multiplication of decimals by 10, 100, and 1000:

- When a decimal is multiplied by 10, move the decimal point one place towards the right.
- When a decimal is multiplied by 100, move the decimal point two places towards the right.
- When a decimal is multiplied by 1000, move the decimal point three places towards the right.

**Example 2: **

### Multiplication of two decimal numbers:

**Example 3:** Multiply 6.25 × 1.2.

**Solution:**

**Exercise:**

1. Find the sum of 3.815 and 327.

2. Add 25 + 0.172.

3. Add 1.8 + 2.16.

4. Subtract 3.12 – 0.8.

5. Find the difference of 15 – 6.108.

6. Subtract 9.501- 9.45.

7. Multiply 1.9 x 0.46.

8. Find the product of 0.4 x 0.17.

9. Multiply 3.42 x 5.15.

10. Multiply 47.12 x 10.

### What have we learned:

• Addition of like and unlike decimal numbers.

• Addition of decimal numbers with whole numbers.

• Addition of decimals by regrouping.

• Estimate to find the sum of decimal numbers.

• Subtraction of decimal numbers by regrouping.

• Subtraction of decimal numbers with whole numbers.

• Estimate to find the difference of decimal numbers.

• Multiplication of decimals with whole numbers.

• Multiplication of decimal numbers by 10,100, and 1000.

• Multiplication of two decimal numbers.

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