3.5: Estimation and Rounding

What is the difference between estimation and rounding? First, let us look at the definitions from Google:

Definition: Estimation

A rough calculation of the value, number, quantity, or extent of something.

Definition: Rounding

Alter (a number) to one less exact but more convenient for calculations.

Now let us look at some examples and determine which one is which:

Figure 3.5.1

Irwin’s example is Estimation. He is estimating an answer by using easier numbers.

Mary’s example is Rounding. She is rounding up 0.76 to 1, to make the numbers easier for her.

One of the main components of Common Core was to make sure that students could perform mental math exercises with ease. Therefore, there is a strong emphasis on percentages, estimation, rounding and multiplication.

Mental Math Percentages

(use these to help you with Example \(\PageIndex{1}\))

10%: Move the decimal once to the left

5%: Find 10% and divide by 2

20%: Find 10% and multiply by 2

1%: Move the decimal twice to the left

Example \(\PageIndex{1}\)

Restaurant Split

For dinner, on Friday night, Chris and Maddy dine at Seven Seas Seafood. They decide to leave their phones in the car, so that there are no distractions and actually have some real conversation! The dinner went very well and the bill comes. Maddy decided to be nice and pay the entire bill; after all, she did order the crab buffet. Now, her phone is in the car and she needs to calculate tip and total. Chris says, “I think you should tip 17%. The service was good, but not great. And do it quick, I want to go home.”

Figure 3.5.2

Solution

Luckily, Maddy remembered how to use her amazing estimation skills and tipped $20.

\[\begin{array}{l}
17 \%=10 \%+5 \%+1 \%+1 \% \\
10 \%=\$ 11.658 \approx \$ 12 \\
5 \% \approx \$ 12 \div 2=\$ 6 \\
1 \%=\$ 1.1658 \approx \$ 1 \\
\$ 12+\$ 6+\$ 1+\$ 1=\$ 20
\end{array} \nonumber \]

Partner Activity 1

Calculate tip for the following restaurant bills using the method presented in Example \(\PageIndex{1}\). Make sure you round to the nearest penny.

1. $85.12
2. $21.47
3. $96.01

Mental Math Multiplication

Example \(\PageIndex{2}\)

Growing up, most of us learned the multiplication tables from zero to ten. Maybe, even some of you reached 11 and 12. But what about the higher numbers? Here is Common Core’s answer.

We need to multiply 18 × 4.

Solution

Figure 3.5.3

We can change the expression as long as we keep it balanced. Dividing 18 by 2 AND multiplying 4 by 2 will keep the expression balanced.

Therefore, 18 × 4 is the same as 9 X 8 = 72.

We do not have to use just two. Any number will work! As long as you keep it balanced!

Partner Activity 2

Here are some advanced multiplication expressions. By yourself, write an easier way to evaluate the expression, so you can use mental math. Then solve. Afterwards, compare your methods to your partner’s method.

1. \(24 \times 3\)
2. \(5 \times 36\)
3. \(16 \times 4\)

Mental Math Subtraction Trick

Example \(\PageIndex{3}\)

One of the hardest things for students to learn is subtracting when the first number (called the minuend) has lots of zeros, like 10,000. There can be confusion with borrowing: which zeros end up being tens and which zeros end up being nines? Common Core has an answer for that, too!

Solution

Let’s try subtracting 1000 - 4983 the traditional way, with borrowing:

\[\begin{array}{cccccc}
& & 9& 9& 9& \\
0&\not{10}&\not{10}&\not{10}&10 \\
\not{1}&\not{0}&\not{0}&\not{0}&\not{0} \\
-&4&9&8&3 \\
\hline
& 5&0&1&7\\
\end{array} \nonumber \]

Some students might see a problem like this and be intimidated by the amount of borrowing. An easier way to subtract: First subtract 1 from both numbers.

Going the Distance

1000 → 9999 and 4983 → 4982

\[\begin{array}{r}
9999 \\
-4982 \\
\hline 5017
\end{array} \nonumber \]

No. More. Borrowing. EVER.

Note

Remember, subtraction is just measuring distance. We are finding the DISTANCE between 10000 and 4983, which is the same DISTANCE as 9999 and 4982

Partner Activity 3

Solve these subtraction problems WITHOUT borrowing! Think about what you should add or subtract to both numbers.

1. 1000 – 679
2. 513 – 284
3. 16452 – 999

Mental Math Addition and Subtraction in Schools Today

Common Core changed the way students add and subtract. Some schools still using carrying (addition) and borrowing (subtraction), like traditional mathematics. However, some schools encourage to teach the Common Core methods explained above, which will lead to mental math abilities, after enough practice.

Practice Problems

1. Estimation
   1. What is 11% of 150?
   2. What is 19% of 400?
2. Rounding
   1. Round 16.456 to the nearest hundredth.
   2. Round $49.347 to the nearest penny.
   3. Round 82.42 to the nearest whole number.
3. Tip
   1. The restaurant bill is $43.18. What is a 15% tip?
   2. The restaurant bill is $56.79. What is a 20% tip?
   3. The restaurant bill is $96.42. What is a 17% tip?
4. Multiply big numbers
   1. Use mental math to multiply \(4 \times 18\).
   2. Use mental math to multiply \(24 \times 3\).
   3. What did you notice about A and B?
5. Subtract from 10000
   1. Use the trick in Example \(\PageIndex{3}\) to subtract 10000 – 9134.
   2. Use the trick in Example \(\PageIndex{3}\) to subtract 100000 – 34678.