# Worksheet: Long Division with Decimals: Placing the Decimal Point

In this worksheet, we will practice explaining the result of multiplying by 10 when dividing with decimals and using it to calculate quotients using long division.

**Q1: **

Calculate , rounding your answer to the nearest hundredth.

**Q2: **

Given that , find the value of .

**Q3: **

James uses the number fact to evaluate . What result does he find?

**Q5: **

Natalie wants to prove, using induction, that is divisible by 5 for all integers .

First, she needs to check the base case when . Substitute into the expression and determine the result when it is divided by 5.

Natalie then makes the assumption that is divisible by 5. She then needs to show that is divisible by 5. To do this, she considers the difference . Write this difference in the form .

- A
- B
- C
- D
- E

At this stage it is not clear whether is divisible by 5. Natalie notices that she may be able to substitute into the expression. By writing as , rewrite the expression for to incorporate .

- A
- B
- C
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- E

Natalie rearranges the equation . She then comes to the following conclusion: If the assumption is correct that the expression is divisible by 5 when , then we have shown that the expression is divisible by 5 when . As we have shown that the expression is divisible by 5 when , we have proved by mathematical induction that the expression is divisible by 5 for all integers .

Is Natalie’s conclusion correct?

- AYes
- BNo

**Q6: **

Liam buys 6.75 kilograms of apples. He is going to make bags of apples weighing 0.75 kilograms each. How many bags does he need?

**Q8: **

A teacher asks Emma and Hannah to calculate .

Which one of them is correct?

- AHannah
- BEmma

**Q9: **

Ethan saves $3.30 a week. He wants to buy a toy that costs $9.90. How many weeks does he need to save until he can buy the toy?

**Q10: **

Calculate the following: