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Interactive Partitioning Cards (2-Digit Numbers)
How to Calculate a Ratio of a Number
- In this example, we are sharing £50 in the ratio 2:3, using steps.
- Step 1 is to find the total number of parts in our ratio.
- 2 + 3 = 5. Our ratio of 2:3 contains 5 parts in total.
- Step 2 is to find the value of one part by dividing the amount by the total number of parts.
- £50 ÷ 5 = £10.
- Step 3 is to scale up the original ratio by the value of one part of the ratio.
- 2:3 becomes £20:£30 when we multiply by £10.
- One person gets £20 and the other gets £30. These values add to our total of £50.
To calculate the ratio of an amount we divide the amount by the total number of parts in the ratio and then multiply this answer by the original ratio.
- We want to work out $20 shared in the ratio of 1:3.
- Step 1 is to work out the total number of parts in the ratio.
- 1 + 3 = 4, so the ratio 1:3 contains 4 parts in total.
- Step 2 is to divide the amount by the total number of parts in the ratio.
- $20 ÷ 4 = $5.
- Each of the four parts of the ratio is worth $5.
- Step 3 is to scale up our original ratio by the value of each part.
- 1:3 multiplied by $5 becomes $5:$15.
- One person gets $5 and the other gets $15.
- We can check our ratio calculation by adding $5 and $15 to get our total amount of $20.
Calculating a Ratio of an Amount: Interactive Questions
Calculating a Ratio of an Amount Worksheets and Answers
How to Calculate a Ratio of a Number
What is a Ratio?
Ratios describe how to share out a given amount and are written with numbers separated by colons. The amount of numbers in the ratio tells us how many groups the quantity is being shared between. The size of each number tells us the proportion of the total amount each group gets.
For example, in the ratio 1:3 there are two different numbers: ‘1’ and ‘3’.
Because there are two numbers, we are sharing an amount between two people.
This ratio means that for every 1 part that the person on the left gets, the person on the right gets 3. The person on the right will have three times as much as the person on the left.
The person on the left will get $5 and the person on the right will get $15. Both numbers add to make the total of $20 but $15 is three times larger than $5. This is why the ratio is 1 to 3.
How to Calculate a Ratio of a Number in Steps
To calculate a ratio of a number, follow these 3 steps:
- Add the parts of the ratio to find the total number of parts.
- Find the value of each part of the ratio by dividing the number by the total number of parts calculated in step 1.
- Multiply each part of the original ratio by the value of each part calculated in step 2.
For example, share $20 in the ratio 1:3.
This means that in this ratio problem, we will be sharing $20 between two people with one person getting three times as much as the other.
We follow the 3 steps above to work out the ratio.
- Step 1: Find the total number of parts
Looking at the ratio 1:3, we have:
1 + 3 = 4
So, we have four parts in total in our ratio.
- Step 2: Divide the amount by the total number of parts
The amount is $20 and the total number of parts is 4.
20 ÷ 4 = 5
Each of the four parts is worth $5.
- Step 3: Multiply each number in the ratio by the value of one part
We have four parts in total and each of these parts are worth $5.
We will multiply both numbers in the ratio 1:3 by $5.
1 x $5 = $5
3 x $5 = $15
The money is shared in the ratio $5:$15.
We have solved our ratio problem to find that one person gets $5 and the other person gets $15.
To check a ratio calculation, add the final values to see if the total is equal to the original amount. If the amounts shared can also be simplified to make the ratio given, the calculation is correct.
$5 + $15 = $20, which is the amount that we started with, so we can be confident that our calculation is correct.
To confirm our ratio calculation, we check that $5:$15 simplifies to give us our original ratio.
Dividing both $5 and $15 by 5, gives us 1:3, which is our original ratio and $15 is indeed three times as much as $5.
Now try our lesson on Sharing in a Ratio Between 3 People where we learn how to extend our steps for calculating ratio to examples where we share an amount between 3 people.
Solving Ratio Problems
- We add the parts of the ratio to find the total number of parts.
- There are 2 + 3 = 5 parts in the ratio in total.
- To find the value of one part we divide the total amount by the total number of parts.
- 50 ÷ 5 = 10.
- We multiply the ratio by the value of each part.
- 2:3 multiplied by 10 gives us 20:30.
- The 50 counters are shared into 20 counters to 30 counters.
Find the value of one part of the ratio by dividing the amount by the total number of parts.
Multiply the ratio by the value of one part.
- 2 + 3 = 5 and so there are 5 parts in the ratio in total.
- We divide by this total number of parts to find the value of each part.
- 50 ÷ 5 = 10.
- We multiply the original ratio by the value of each part.
- We have 20:30.
How to Solve Ratio Problems
Here is an example of a ratio problem. In this lesson we will look at how we solve ratio problems using a few examples.
We are asked to share £50 in the ratio 2:3.
This means that for every two parts a person receives, a second person is given three parts.
We begin by finding the total number of parts.
There are two parts highlighted in green and three parts highlighted in orange.
We have five parts in our ratio in total.
In total, all five parts are worth £50.
To be able to find out how much money each person is given, we need to know how much one part is worth.
Each part in the ratio must have the same value.
This means that £50 must be shared equally over the five parts.
To share equally, we divide £50 by 5.
50 ÷ 5 = 10
So, each part is worth £10.
Now that we know that one part is worth £10, we can find out what two parts and three parts are worth.
If one part is £10, two parts must equal £20.
£10 x 2 = £20
If one part is £10, three parts must equal £30.
£10 x 3 = £30
The money in this ratio problem has been shared £20:£30.
One person receives £20 and the other receives £30. We can check our answer by adding the two amounts to see if they make our original total.
£20 + £30 = £50
Now try our lesson on How to Calculate a Ratio of a Number where we break down the process of sharing in a ratio into three steps.
Multiplication Using Factors
- Factors are numbers that divide exactly into other numbers.
- Larger numbers can be written as their factors multiplied together.
- For example, 15 = 3 × 5. 3 and 5 are factors of 15.
- Instead of multiplying by 15, we can multiply by 3 and then by 5.
- 3 × 15 is the same as 3 × 3 × 5.
- 3 × 3 = 9 and then 9 × 5 = 45.
- Therefore, 3 × 15 = 45.
- This method is used to multiply by larger numbers.
- Multiplying by smaller numbers in stages is easier than multiplying by larger numbers in one go.
To multiply by a larger number, multiply by its factors.
- 24 can be written as the product of its two factors, 4 × 6.
- 5 × 24 can be written as 5 × 4 × 6.
- 5 × 4 = 20 and 20 × 6 = 120.
- Therefore, 5 × 24 = 120.
Multiplication Using Factors: Interactive Questions
Multiplication Using Factors: Interactive Questions
Multiplication Using Factors Worksheets and Answers
Multiplication Using Factors
How to Multiply Numbers Using Factors
A factor is a number that divides exactly into another number. Instead of multiplying by a larger number in one go, it can be easier to multiply by the factors that multiply to make this number. For example, to multiply 3 × 70, it can be easier to multiply 3 × 7 and then multiply by 10.
Choose two factors of the larger number which will be easiest to multiply by. For example, multiplying 3 × 7 × 10 is easier than multiplying 3 × 2 × 35. Even though 2 × 35 = 70, 35 is not an easy number to multiply by.
3 × 7 = 21 and this is a known fact from our times tables. It is then simple to multiply by 10 by placing a 0 digit on the end of 21.
3 × 7 × 10 = 210 and so, 3 × 70 = 210.
Here is 5 × 6.
We can write 6 as 2 × 3.
5 × 6 can be written as 5 × 2 × 3.
5 × 2 = 10 and then 10 × 3 = 30.
5 × 2 × 3 = 30 and so, 5 × 6 = 30.
Factors can be used to help us work out any multiplication. This method can also be used for teaching and learning the times tables. If you knew the 2 and 3 times tables but were still learning the 6 times tables, you can multiply 5 by 2 and then by 3 to work out 5 × 6.
We choose to do 5 × 2 × 3 rather than 5 × 3 × 2 because 5 × 2 = 10 and this is an easy number to multiply by. When teaching multiplication by using factors, try to find groups of easier multiplications.
Here is 7 × 9.
9 = 3 × 3 and so, multiplying by 9 is the same as multiplying by 3 and then by 3 again.
7 × 9 = 7 × 3 × 3.
7 × 3 = 21 and 21 × 3 = 63. We can multiply 21 by 3 by multiplying each digit in 21 by 3.
2 tens × 3 = 6 tens and 1 × 3 = 3. 21 × 3 = 63.
7 × 3 × 3 = 63 and so, 7 × 9 = 63.
Multiplying by 3 and then 3 again can be used as a strategy to multiply a number by 9.
How to Multiply Larger Numbers
It is easier to multiply by factors of a larger number than it is to multiply by the larger number directly. Find two numbers that multiply to make the larger number and then multiply by these two numbers separately instead.
For example, here is 8 × 14.
14 = 7 × 2. We can write 8 × 14 as 8 × 7 × 2.
8 × 7 = 56 and then 56 × 2 = 112. We can work out 56 × 2 because it is doubling. Doubling means to add the number to itself. 56 + 56 = 112 because 50 + 50 = 100 and 6 + 6 = 12.
8 × 7 × 2 = 112 and so, 8 × 14 = 112.
Notice that we chose to multiply 8 × 7 × 2 rather than 8 × 2 × 7.
This is because 8 × 2 × 7 leads to 16 × 7, which is not an easy multiplication. With 8 × 7 × 2, we used a times table fact for 8 × 7 and then only had to double this answer.
Now try our lesson on Multiplying by Multiples of 10 and 100 where we learn how to multiply by numbers in the 10 and 100 times table.
Mental Addition of 2-Digit Numbers
- To add 2-digit numbers mentally, it is important to know the addition results shown above.
- To add two numbers mentally, add the digits separately.
- Add the tens digits and then look at the ones digits to decide what the answer ends in.
- For example in 32 + 29, we add the tens digits first.
- 3 + 2 = 5.
- Adding the ones, 2 + 9 = 11.
- Because 11 ends in a 1, we know our answer ends in a 1.
- We increase the 5 in the tens column to a 6.
- 32 + 29 = 61.
Add the tens column and then the ones column separately.
The answer ends in the same digit that the sum of the units column ends in.
- We first add the tens column digits.
- 3 + 4 = 7. We hold this number in our head.
- We now add the digits in the ones column.
- 6 + 9 = 15.
- So we add one to the tens column and our answer ends in a 5.
- The tens column increases from 7 to 8 and the ones column is 5.
- 36 + 49 = 85.
Mental Addition of 2-Digit Numbers: Interactive Questions
Mental Addition of 2-Digit Numbers: Interactive Questions
Mental Addition of 2-Digit Numbers Worksheets and Answers
Mental Addition
What is Mental Addition?
Mental addition means to add numbers in your head without writing down any steps or notes as part of the calculation. All of the calculations are done in your head.
There are various strategies for mental addition, however, with each method, it is important to hold values in your head. The fewer the steps, the better the method as it is easier to hold the numbers in your head.
How to do Mental Addition
To do mental addition, use the following steps:
- Add the digits in the tens column together and remember this result.
- Add the digits in the units column together.
- If this result is larger than ten then add one to your tens column answer from before.
- The units column of the units column result is the units digit of your answer.
We only hold the result of the tens column in our head while we add the units column.
To add 2-digit numbers, it is first worth knowing the following 1-digit additions:
In this first example, we have 32 + 29.
The first step is to add the digits in the tens column.
3 + 2 = 5.
The next step is to add the digits in the units column.
2 + 9 = 11
11 is larger than 10 and so, we need to add 1 to our result from adding the tens column. This is because adding 11 is the same as adding ten and then adding 1.
The 5 in the tens column becomes a 6.
Because 11 ends in a 1, we know that our digit in the units column of our answer is also a 1.
32 + 29 = 61
In this next example we have the mental addition of the 2-digit numbers 44 + 28.
The first step to adding two 2-digit numbers mentally is to add the digits in the tens column.
4 + 2 = 6
The next step is to add the digits in the units column.
4 + 8 = 12
To add 12, we add 10 and 2.
This means that we increase the tens column by 1 and our answer ends in a 2.
The 6 increases to a 7.
The answer ends in a 2 because 12 ends in a 2.
44 + 28 = 72
When teaching mental addition, we recommend trying to hold the first result in your head whilst carrying out the next part of the calculation. When first learning mental addition, start by writing the result a few times until you get the hang of the method.
Once you get better at performing the written method you will find that it is easier to remember the steps and it will be easier to hold numbers in your head.
In this example of mental addition we have 28 + 25.
We will try and speed up our method.
We add the tens:
2 + 2 = 4
8 + 5 = 13 and so, we increase 4 to 5 in the tens column. Our answer ends in 3.
28 + 25 = 53
Here is another example of adding 2-digit numbers mentally.
We have 36 + 49.
Adding the tens digits we have 3 + 4 = 7.
Adding the units digits we have 6 + 9 = 15.
We increase the 7 to an 8 and our answer must end in a 5.
36 + 49 = 85
In this example of adding mentally, we have 69 + 29.
Adding the tens, 6 + 2 = 8.
Adding the units, 9 + 9 = 18.
Therefore, we increase the 8 in the tens column to a 9 and our answer ends in an 8.
69 + 29 = 98
With practice, this method is one of the quickest ways to add 2-digit numbers mentally.
When teaching mental addition, if your child struggles with the process, it is worth focusing on writing the steps down until they are mastered. This will make it easier to transfer the process to holding numbers in their head.
Now try our lesson on 2-Digit Column Addition where we learn how to use the column addition method.
