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 7.367÷73.6, rounding your answer to the nearest hundredth.

Q2:

Given that 6,141÷69=89, find the value of 6.141÷0.089.

Q3:

James uses the number fact 63÷7=9 to evaluate 0.63÷0.07. What result does he find?

Q4:

A group of cats weigh 45.5 kilograms. Assuming that each cat weighs 4.55 kilograms on average, determine the number of cats in the group.

Q5:

Natalie wants to prove, using induction, that 𝑓(𝑛)=23 is divisible by 5 for all integers 𝑛1.

First, she needs to check the base case when 𝑛=1. Substitute 𝑛=1 into the expression and determine the result when it is divided by 5.

Natalie then makes the assumption that 𝑓(𝑘)=23 is divisible by 5. She then needs to show that 𝑓(𝑘+1)=23() is divisible by 5. To do this, she considers the difference 𝑓(𝑘+1)𝑓(𝑘). Write this difference in the form 𝑎2𝑏3.

  • A 𝑓 ( 𝑘 + 1 ) 𝑓 ( 𝑘 ) = 7 2 2 3
  • B 𝑓 ( 𝑘 + 1 ) 𝑓 ( 𝑘 ) = 9 2 4 3
  • C 𝑓 ( 𝑘 + 1 ) 𝑓 ( 𝑘 ) = 7 2 4 3
  • D 𝑓 ( 𝑘 + 1 ) 𝑓 ( 𝑘 ) = 9 2 2 3
  • E 𝑓 ( 𝑘 + 1 ) 𝑓 ( 𝑘 ) = 4 2 4 3

At this stage it is not clear whether 𝑓(𝑘+1) is divisible by 5. Natalie notices that she may be able to substitute 𝑓(𝑘) into the expression. By writing 72 as 52+22, rewrite the expression for 𝑓(𝑘+1)𝑓(𝑘) to incorporate 𝑓(𝑘).

  • A 𝑓 ( 𝑘 + 1 ) 𝑓 ( 𝑘 ) = 2 + 2 𝑓 ( 𝑘 )
  • B 𝑓 ( 𝑘 + 1 ) 𝑓 ( 𝑘 ) = 5 2 + 2 𝑓 ( 𝑘 )
  • C 𝑓 ( 𝑘 + 1 ) 𝑓 ( 𝑘 ) = 5 2 + 2 𝑓 ( 𝑘 )
  • D 𝑓 ( 𝑘 + 1 ) 𝑓 ( 𝑘 ) = 5 2 + 𝑓 ( 𝑘 )
  • E 𝑓 ( 𝑘 + 1 ) 𝑓 ( 𝑘 ) = 5 2 + 𝑓 ( 𝑘 )

Natalie rearranges the equation 𝑓(𝑘+1)=52+3𝑓(𝑘). She then comes to the following conclusion: If the assumption is correct that the expression is divisible by 5 when 𝑛=𝑘, then we have shown that the expression is divisible by 5 when 𝑛=𝑘+1. As we have shown that the expression is divisible by 5 when 𝑛=1, we have proved by mathematical induction that the expression is divisible by 5 for all integers 𝑛1.

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?

Q7:

Mason plays guitar 1.2 hours a day. If he played for a total of 10.8 hours, how many days did he play for?

Q8:

A teacher asks Emma and Hannah to calculate 0.6÷0.01.

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:

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