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Three division signs | Partition division| Quotition division | Two digit division with no remainders | Three or more digit division with no remainders | Division with remainders | Division with numbers containing zeros | Algorithm based on repeated subtraction| Estimation and mental division | Can we divide by zero? | Quick quiz| Mixed operations quiz | Three symbols are commonly used to mean 'division', they are shown in the table below.
All division problems can be expressed using any of these signs, but when we actually try to solve a division problem using the long or short division algorithm,
we use
Knowing whether a problem is partition or quotition division can help us picture and solve the problem. In partition division a number is separated into a specified number of equal sized groups and we want to find the number in each of these equal sized groups. Example 1: 28 people need to get to a wedding by car and there are 7 cars available. If each car carries the same number of people, how many people will travel in each car to get everyone to the wedding? We have 28 people who need to fit into 7 cars, so the question becomes, 28 divided by 7. 28 people ÷ 7 cars = 4 people per car 28 divided by 7 can be thought of as 28 shared among 7. Thinking of division as sharing can help us picture and then solve partition problems.
28 people shared among 7 cars is 4 people in each car.
In quotition division we know the number in each group and we need to find out the number of groups. Example 2: 28 people need to get to a wedding by car and 4 people can fit into a car. How many cars are needed to get everyone to the wedding? We can fit 4 people into each car, so the question becomes, 28 divided by 4. 28 people ÷ 4 people per car = 7 cars needed How many groups of 4 people are there in 28 people?
There are 7 groups of 4 people in 28 people. We can see that 28 divided by 4 is 7 because 7 x 4 = 28
So we can clearly see that,
Example 3: Using the long division algorithm, 72 ÷ 3 = 24
Example 4: Full explanation of the short division algorithm, 96 ÷ 8 = 12
Example 5: Using the long division algorithm, 624 ÷ 4 = 156
We can also use the short division algorithm. We use the same ideas as for long division, there is just less recording. View the movies to see the difference. Example 6: Full explanation using the short division algorithm, 536 ÷ 2 = 268
Example 7: Full explanation using the long division algorithm, 1224 ÷ 36 = 34
When a number does not divide exactly into another number, we say that there is a remainder - something left over. 3 ways to express remainders We can express remainders in three different ways. Which way we choose to use depends on the context of the question and answer.
Example 8: 34 ÷ 5 = 6 remainder 4
Example 9: Full explanation of long division with decimal remainder, 146 ÷ 16 = 9.125 NOTE: In his explanation, the narrator mistakenly says 'we can think of the 2 as 2 tenths' - what he meant to say was 'we can think of the 2 as 20 tenths'!
Example 10: Expert performance (with an error!) of short division algorithm, 613 ÷ 25 = 24.52
To solve division problems which involve numbers containing zeros, we use exactly the same procedure as for any other numbers. The only reason we make special mention of these sorts of numbers is because people often make mistakes! Example 11: Full explanation of short division focusing on recording zeros, 415 ÷ 2 = 207.5
Example 12: Expert performance of 7061 ÷ 23 = 307
Another algorithm (sometimes irreverently called "dribble down the side") uses the fact that division can be interpreted as repeated subtraction. This is an alternative algorithm which is easier for children. However, it fails for division with decimals, where the standard algorithm works with only minor changes. Example 13: Using the estimation method for 900 ÷ 24
The setting out required in this method is obviously much longer than for standard long division approach; however this method does not require us to have all our multiplication facts at our finger tips! We can use one multiplication fact and subtract the same number repeatedly until we get down to small numbers.
Estimation and mental division You probably make mental division estimations more often than you realise! Do these situations seem familiar? "I've got $4.90, can I afford a cup of coffee for Marie and me if a coffee costs $2.40?" "The bill is $135.80 and there are 6 of us, so let's split the bill evenly." Read the description below to see how one group of people splits the bill at a restaurant. Example 14: Splitting the bill "We'll round the cost to $136 and divide the bill evenly. So 136 divided by 6 is ... well 6 x 20 = 120, so we owe $20 each so far, then we have $16 left over. 6 x 3 = 18, so that means 20 + 3 dollars each would cover the bill with a $2 left over for a tip! That tip is not much!"
People often get confused about whether we can divide a number by zero and if not why not. A simple example reveals all! Here are three explanations: based on quotition, partition and division as the inverse of multiplication. There is no answer because the question is meaningless. (Remember that there is NO ANSWER - the answer is not zero) Example 15: 8 ÷ 0 How many groups of 0 things can I make from 8 things? I start with 8, and take away zero, then take away another zero, and another and another..... It is not possible to answer this question. 8 ÷ 0 is not a meaningful question. We can also think of 8 ÷ 0 as 8 things shared among no people (partition interpretation) but that is rather weird. You might think that the answer could be 8 but we can use multiplication to show why it is not. We know that multiplication and division are the inverse of one another, so, if 8 ÷ 0 = 8 then it follows that 8 x 0 = 8. But what does 8 x 0 equal? 0. We have shown that there is no answer to 8 ÷ 0 since 0 times any number will be zero, and hence never 8.
Further questions: 1. Share $4008 dollars equally among 6 people. 2. There are 208 apples. How many people can get 8 apples each? 3. When two numbers are multiplied the answer is 765. If one of the numbers is 17, what is the other number? 4. Jenny wanted to divide a certain number by fourteen. However, she pressed the multiplication button on her calculator instead of the division button and got the answer, 54 096. What was the correct answer? 5. How many groups of 4 students can be made if there are 52 students in the drama class? 6. Divide 9574 by 23 7. The area of Tasmania is 70 000 km2 and the area of the Australian Capital Territory is 2500 km2. How many times larger than the Australian Capital Territory is Tasmania? 8. Water empties from a tank at a rate of 250 litres per minute. How long will it take for the full tank of 13 500 litres to empty? 9. A photocopier produces 100 copies per minute. There is an order for 25 000 copies. Can the photocopier complete this order in 4 hours? Give reasons for you answer. 10. Four blocks of wood are laid down in a row with a space of 8 cm between them. If the distance from the start of the first block to the end of the last block is 148 cm, how long was each block of wood? To view the quiz answers, click here.
1. Vincent had a certain number of eggs to sell. He sold 8 eggs and then his hens laid another 15 eggs. He dropped 14 eggs and sold another 12 eggs. If he now has 20 eggs left, how many eggs were there at the beginning? 2. Angela and Xuping wish to share 36 sweets. If Xuping is to get twice as many sweets as Angela, how many sweets will each girl get? 3. At my birthday party there were 56 green balloons, 27 red balloons and the rest of the balloons were yellow. There were 142 balloons altogether. (a) How many yellow balloons were there? (b) There were 12 people at my party. Each person was given 2 of the red balloons, 3 of the yellow balloons and 4 of the green balloons to take home. How many balloons were left? 4. Claudine’s step is 58 cm. How many steps will she need to take to walk 11 020 cm? 5. What number do you get if you multiply 35 by 8, divide by 10 and then add 602? 6. Find the difference between the sum of 12 and 18 and the product of these two numbers. 7. I have 9 rows of jelly beans with 5 jelly beans in a row and 4 rows of smarties with 12 smarties in a row.(a) Which sweet do I have more of? (b) How many more do I have of this sweet? 8. Harry is 6 years of age. His mother is five times his age and his father is six times his age. Harry has worked out that if he multiplies his brother Bob’s age by 3 this number would still be 3 years younger than his mother’s age . (a) How old is Bob? (b) What is the total of the ages of the four people in this family? 9. The final score for a football match is 193. There are six points for a goal and 1 point for a behind. If there were 19 behinds in the match how many goals were scored? 10. A can of cat food has a mass of 125g and a can of dog food has a mass of 175g. Scott buys 24 cans of cat food and 16 cans of dog food. (a) What is the total mass of the food bought? (b) If the dog eats the same amount of food each day and it takes 2 weeks to eat the dog food just bought, how much does it eat each day? 11. Seventeen bags each contain the same number of videos. The total number of videos could be:
12. One shelf has five times as many books as another shelf. There are 85 books on the shelf with the larger number of books. How many of these books would need to be moved to the other shelf so that each shelf contained an equal number of books. 13. From the digits 1, 6, 8, 9, select two 2-digit numbers so that their product will be the largest possible. No digit can be used more than once. 14. In a certain year level there are 218 students. The teacher wants the students, in groups of no more than 6, to discuss which charity should receive the money they have collected during their class fund raising. What is the minimum number of groups that the class will need to form? 15.
The above table shows the number of frequent flyer points it takes for a return trip from Melbourne to various Australian cities, flying economy class. Kylie needs to take two trips to Brisbane and two to Hobart in the next quarter using her points and she also wants to take a holiday to either Sydney or Adelaide using frequent flyer points. She has 46655 points and she always travels economy class. Has she enough points to do this? Explain. 16. A yoga class costs $12 per lesson. There are 10 lessons in a term and 4 terms in a year. How much would a person save if she bought a yearly subscription to these classes for $418? 17. For a school trip, 990 children are to travel on 24 buses. The teachers put an equal number of children on each bus and then found that they had some children left over. (a) How many children were left over? (b) Each of the children left over was put as one extra onto the buses, so that some buses now had one extra student. How many children were on each bus that carried an extra student? 18. Which would cost more and by how much? Twelve books at $23 each or 16 CDs at $21 each? 19. What is the smallest positive two-digit number that leaves a remainder of 2 when it is divided by 5, and a remainder of 1 when it is divided by 6? 20. Ming has a certain number of birds and a certain number of cages. If three birds are placed in each cage there are 2 birds left over. Later, the birds are moved and one cage is left empty, one cage has only one bird in it and 4 birds are placed in each of the other cages. How many birds and how many cages are there? To view the mixed operations quiz answers, click here.
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