2. Rational numbers

Lesson

When we add whole numbers, such as $82+35$82+35, we use place value to help us solve the addition and ensure our answer is correct.

We start by adding the values in the ones column, remembering to carry in to the tens column where necessary, and then by adding the values in the tens column.

Hundreds | Tens | Ones | |

$8$8 | $2$2 | ||

$+$+ | $3$3 | $5$5 | |

$1$1 | $1$1 | $7$7 |

We can see when we added the values in the tens column, we carry over into the hundreds column, giving us a final answer of $117$117.

When adding numbers with decimals we can use exactly the same approach.

If we were adding $8.2$8.2 and $3.5$3.5 we can use a place value table again, only this time we need to include place values to the right of the decimal place. We also need to include the decimal point as it's own column between the ones and the tenths.

Tens | Ones | $\cdot$· | Tenths | |

$8$8 | $\cdot$· | $2$2 | ||

$+$+ | $3$3 | $\cdot$· | $5$5 | |

$1$1 | $1$1 | $\cdot$· | $7$7 |

In this case, when we added the values in the ones column, we carry over into the tens column, giving us a final answer of $11.7$11.7.

Remember!

Adding decimals uses the same process as we use for whole numbers, but we need to consider the value of the digits and where our decimal point needs to be.

As our numbers get bigger, and contain more decimal places, we just follow the same simple procedure. The videos below take you through the process for much larger numbers with place values ranging from *thousands* to *thousandths*.

In the second video, you can see an example of adding two numbers with decimals, but this time we need to regroup, or rename, some decimals. We use the same process that we use for whole numbers, so your prior knowledge, as well as the first video, will help here.

Careful!

Be sure to line up the numbers, so that the decimal points are in line, as this makes sure that the numbers with the same place value are in the same place column.

Find $3.8+4.6$3.8+4.6 giving your answer as a decimal.

Evaluate $644.969+229.713$644.969+229.713

When we subtract numbers with decimals, we again follow the same process as working with whole numbers and use place values.

What is $87-5.45$87−5.45?

**Think: **If we wanted to take away $5.45$5.45 from $87$87 we can use a place value table. But what do we put after the decimal point for $87$87?

Whole numbers don't have any written decimal values, so it helps to think of them as having zeros after the decimal point.

It also helps to have an rough estimate of our expected answer, so we know our calculation is correct. Here, $5.45$5.45 is a little bit *less *than $6$6, so we would expect our answer to be a little bit *more *than $87-6=81$87−6=81.

**Do:** We can rewrite $87-5.45$87−5.45 as $87.00-5.45$87.00−5.45, and use the vertical algorithm to solve the subtraction.

Tens | Ones | $\cdot$· | Tenths | Hundredths | |

$8$8 | $7$7 | $\cdot$· | $0$0 | $0$0 | |

$-$− | $5$5 | $\cdot$· | $4$4 | $5$5 | |

$8$8 | $1$1 | $\cdot$· | $5$5 | $5$5 |

**Reflect: **By adding trailing zeros to our whole number, we we were able to subtract a number with decimals from it. Our calculated answer of $81.55$81.55 fits with our expectation that the answer would be a little bit more than $81$81.

Now, let's watch a video to see how using decimals with values in the tenths and hundredths places.

In our second video, we are going to work through a subtraction where we need to regroup. When we are subtracting using a place value system (e.g. a vertical algorithm) and need to take a larger number away from a smaller number, we need to exchange a value from the previous place. For example, we could exchange $1$1 tenth for $10$10 hundredths. Subtracting decimal values uses the same process as with whole numbers, only now there are values in the columns after the decimal point. Let's take a look at this process in more detail.

Remember!

We always start our calculations from the smallest place value column.

Find $5.9-1.7$5.9−1.7. Write your answer in decimal form.

Evaluate $64.51-53.47$64.51−53.47

Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.