1.6: Estimation of Whole Number Multiplication and Division
Remember Jonah and Sarah and the penguin population? Well now that the two have calculated the population, they are moving on to food.
Jonah went to meet his friend Sarah for lunch. Sarah is also a zoo volunteer. She has been working in the penguin arena. There are 57 penguins at the city zoo. One of her jobs is to feed the penguins.
“Wow, what a morning. I had to figure out how much seafood to order for the seals. My whole morning has been solving problems. I thought I left math at school,” Jonah says, biting into his peanut butter sandwich.
“What’s so hard about that? I could figure out how much to order for the penguins without even using a piece of paper,” Sarah states.
“What! How can you do that?”
“Estimation. The penguins eat about 18,000 fish per month,” Sarah says, biting into her sandwich.
“18,000 fish!! How do you know that?”
“I told you, estimation. There are 57 penguins who each eat about 8 – 10 fish per day. You don’t need an exact number, just be sure to have enough fish. Once you know that, the rest is easy,” Sarah smiles and takes a sip of her water.
Jonah is completely perplexed.
How did Sarah do that so quickly? What is estimation all about anyway? Could he have used estimation to solve his own problem? You will learn all that you need to know to help Jonah to understand how Sarah figured out the penguin food so quickly by reading through this next lesson. Pay close attention. At the end of this Concept, we’ll revisit this problem and see how she did it.
Guidance
In the last Concept, you learned how to estimate sums and differences using rounding. Sums and differences refer to addition and subtraction. But what about products and quotients? Those are vocabulary words from the first concept.
Let’s review what they mean before we continue.
A product is the answer to a multiplication problem. A quotient is the answer to a division problem.
How do we estimate a product? We can estimate the product of a multiplication problem by rounding the factors that we are multiplying. We use the same rounding rule as with sums and differences.
\begin{align*}12 \times 19 = \underline{\;\;\;\;\;\;\;}\end{align*}
Let’s estimate by rounding each factor to the nearest ten.
\begin{align*}& 12 \ \text{rounds to} \ 10\\ & 19 \ \text{rounds to} \ 20\\ & 10 \times 20 = 200\end{align*}
Our estimate is 200.
This may seem a little harder than adding and subtracting, but you should be able to use mental math to estimate each product.
We can estimate a quotient in the same way.
\begin{align*}32 \div 11 =\underline{\;\;\;\;\;\;\;}\end{align*}
Let’s estimate by rounding each value to the nearest tenth.
\begin{align*}& 32 \ \text{rounds to} \ 30\\ & 11 \ \text{rounds to} \ 10\\ & 30 \div 10 = 3\end{align*}
Our estimate is 3.
Sometimes, when working with division, we need to find a compatible number, not just a rounded number. What is a compatible number? A compatible number is one that is easily divisible.
Let’s look at one that uses compatible numbers.
\begin{align*}2321 \div 8 = \underline{\;\;\;\;\;\;\;}\end{align*}
This one is tricky. Normally, we would round 2321 to 2300, but 2300 is not easily divisible by 8. However, 2400 is easily divisible by 8 because 24 divided by 8 is 3.
2400 is a compatible number.
Let’s round and estimate.
\begin{align*}& 2321 \ \text{becomes the compatible number} \ 2400\\ & 8 \ \text{stays the same}\\ & 2400 \div 8 = 300\end{align*}
Our estimate is 300.
Sometimes, it can be a little tricky figuring out whether you should round or use a compatible number. You have to do what you think makes the most sense.
Now let's practice with a few examples.
Example A
\begin{align*}34 \times 18 =\underline{\;\;\;\;\;\;\;}\end{align*}
Solution: 30 x 20 = 600
Example B
\begin{align*}187 \times 11 = \underline{\;\;\;\;\;\;\;}\end{align*}
Solution: 200 x 10 = 2000
Example C
\begin{align*}120 \div 11 = \underline{\;\;\;\;\;\;\;}\end{align*}
Solution:120 /div 10 = 12
Now we can apply what we have learned about estimation to our real world problem.
Let’s go back to Sarah and Jonah having lunch. Here is a review of the conversation that they had.
“Wow, what a morning. I had to figure out how much seafood to order for the seals. My whole morning has been solving problems. I thought I left math at school,” Jonah says, biting into his peanut butter sandwich.
“What’s so hard about that? I could figure out how much to order for the penguins without even using a piece of paper,” Sarah states.
“What! How can you do that?”
“Estimation. The penguins eat about 18,000 fish per month,” Sarah says, biting into her sandwich.
“18,000 fish!! How do you know that?”
“I told you, estimation. There are 57 penguins who each eat about 8 – 10 fish per day. You don’t need an exact number, just be sure to have enough fish. Once you know that, the rest is easy,” Sarah smiles and takes a sip of her water.
Let's pause here for a minute and underline any important information. This has been done for you in the paragraph above. Sarah claims that she can estimate to figure out how much fish the penguins eat. Sarah begins by saying that the penguins eat about 18,000 fish per month.
Now that we know all about estimation, let’s look at how she used estimation to come up with this number by learning some more of the story. Jonah sat puzzled for a long time. Then he finally gave up. “Okay, I give up. How did you figure it out?” he asked. “There are 57 penguins in the pen. I began by rounding 57 to 60 because 57 is a tough number to work with,” Sarah said smiling. “The penguins each eat 8 to 10 fish per day. Well, 10 is a much easier number to work with than 8, so I rounded up to 10.” “If there are 60 penguins, each eating 10 fish per day-that is 600 fish per day. I estimated that product by multiplying in my head.” “There are 30 days in a month. So I estimated 600 per day times 30 days. My final answer is 18,000 fish.” Sarah looked at Jonah, whose mouth was open. Then he smiled.
“That’s great for you,” he said. “But that wouldn’t have worked for my problem. I needed a closer answer. I would have ended up with way too much seafood.”
Is he correct? Let’s take a look.
Here is what the math looked like in Sarah's problem.
\begin{align*}& 57 \ \text{penguins rounded to} \ 60 \ \text{penguins}\\ & 8 - 10 \ \text{fish rounded to} \ 10 \ \text{fish}\\ & 60 \times 10 = 600 \ \text{fish per day}\\ & 30 \ \text{days in one month}\\ & 600 \times 30 \ \text{days} = 18,000 \ \text{fish}\end{align*}
Sarah’s answer makes sense. She did not need an exact answer, so this was the perfect opportunity to use estimation. What about Jonah? Would estimation have worked for his problem? Let’s revisit it. Here are the facts.
There are 43 seals at the zoo. Each seal eats 11 lbs of seafood per day. How many 25 lb buckets does Jonah need to order? We can estimate to find our answer.
43 rounds to 50-if we round down some seals won’t eat
11 rounds to 10
\begin{align*}50 \times 10 = 500 \end{align*} pounds per day
\begin{align*}500 \times 7 = 3500 \end{align*} pounds per week
\begin{align*}3500 \div 25\end{align*} lbs per bucket \begin{align*}= 140\end{align*} buckets
Jonah figured out using exact math that he needed to order 133 buckets of seafood. Estimating, Jonah would have ordered 140 buckets.
\begin{align*}140 - 133 = 7\end{align*} buckets \begin{align*}\times \ 25\end{align*} pounds of fish \begin{align*}= 175\end{align*} extra pounds of fish
That would have been a lot more seafood than he would have needed. This is an example of the key things to think about when estimating:
- The answer must make sense for the problem.
- It must be reasonable.
- We need an answer that is close to the exact answer.
- If the answer does not make sense, then we have to use exact math.
Vocabulary
Here are the vocabulary words used in this Concept.
- Round
- to change a number to the nearest ten, hundred or thousand etc.
- Product
- the answer to a multiplication problem
- Quotient
- the answer to a division problem
- Factors
- the numbers being multiplied in a problem
- Compatible number
- a number that is easily divisible by the divisor in an estimation problem.
Guided Practice
Here's one for you to try on your own.
\begin{align*}869 \div 321 = \underline{\;\;\;\;\;\;\;}\end{align*}
Answer
Let’s estimate by rounding each value to the nearest hundred.
\begin{align*}& 869 \ \text{rounds to} \ 900\\ & 321 \ \text{rounds to} \ 300\\ & 900 \div 300 = 3\end{align*}
Our estimate is 3.
Video Review
Here is a video that you can use to review this concept.
James Sousa on Estimating Solutions to Multiplication and Division
Practice
Directions: Use what you have learned to estimate the following products and quotients.
1. \begin{align*}17 \times 12 = \underline{\;\;\;\;\;\;\;}\end{align*}
2. \begin{align*}22 \times 18 = \underline{\;\;\;\;\;\;\;}\end{align*}
3. \begin{align*}9 \times 18 = \underline{\;\;\;\;\;\;\;}\end{align*}
4. \begin{align*}7 \times 23 = \underline{\;\;\;\;\;\;\;}\end{align*}
5. \begin{align*}36 \times 40 = \underline{\;\;\;\;\;\;\;}\end{align*}
6. \begin{align*}13 \times 31 = \underline{\;\;\;\;\;\;\;}\end{align*}
7. \begin{align*}9 \times 27 = \underline{\;\;\;\;\;\;\;}\end{align*}
8. \begin{align*}11 \times 32 = \underline{\;\;\;\;\;\;\;}\end{align*}
9. \begin{align*}19 \times 33 = \underline{\;\;\;\;\;\;\;}\end{align*}
10. \begin{align*}22 \times 50 = \underline{\;\;\;\;\;\;\;}\end{align*}
11. \begin{align*}43 \div 6 = \underline{\;\;\;\;\;\;\;}\end{align*}
12. \begin{align*}19 \div 10 = \underline{\;\;\;\;\;\;\;}\end{align*}
13. \begin{align*}44 \div 8 = \underline{\;\;\;\;\;\;\;}\end{align*}
14. \begin{align*}72 \div 7 = \underline{\;\;\;\;\;\;\;}\end{align*}
15. \begin{align*}17 \div 8 = \underline{\;\;\;\;\;\;\;}\end{align*}
16. \begin{align*}43 \div 9 = \underline{\;\;\;\;\;\;\;}\end{align*}
17. \begin{align*}62 \div 8 = \underline{\;\;\;\;\;\;\;}\end{align*}
18. \begin{align*}102 \div 18 = \underline{\;\;\;\;\;\;\;}\end{align*}
19. \begin{align*}395 \div 11 = \underline{\;\;\;\;\;\;\;}\end{align*}
20. \begin{align*}778 \div 22 = \underline{\;\;\;\;\;\;\;}\end{align*}
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Compatible number
A number that is easily divisible by the divisor in an estimation problem.Estimate
To estimate is to find an approximate answer that is reasonable or makes sense given the problem.Factors
Factors are numbers or values multiplied to equal a product.Product
The product is the result after two amounts have been multiplied.Quotient
The quotient is the result after two amounts have been divided.Image Attributions
Here you'll learn to estimate products and quotients of whole numbers using rounding.