Unit 5 Decimal Operations — Unit Plan

Base-ten diagrams can help us understand subtraction. Suppose we are finding $0.23 - 0.07$. Here is a diagram showing 0.23, or 2 tenths and 3 hundredths.

Subtracting 7 hundredths means removing 7 small squares, but we do ;not have enough to remove. Because 1 tenth is equal to 10 hundredths, we can decompose one of the tenths (1 rectangle) into 10 hundredths (10 small squares).

Base ten diagram. 0 point 23. Two rectangles in the tenths column. 3 small squares in the hundredths column. A dotted rectangle is drawn around one of the rectangles with an arrow to 10 small squares. The arrow is labeled decompose.

We now have 1 tenth and 13 hundredths, from which we can remove 7 hundredths.

Base ten diagram. 0 point 23. One rectangle in the tenths column. 13 small squares in the hundredths column. 7 small squares have an X through them. The words subtract 0 point 0 7 is below the small squares.

We have 1 tenth and 6 hundredths remaining, so $0.23 - 0.07 = 0.16$.

Here is a vertical calculation of $0.23 - 0.07$.

Vertical subtraction. First line. 0 point 23. The 2 is crossed out and has a 1 above it. The 3 is crossed out and has 13 above it. Second line. Minus 0 point 0 0 7. Horizontal line. Third line. 0 point 17.

Notice how this representation also shows that a tenth is decomposed into 10 hundredths in order to subtract 7 hundredths.

This works for any decimal place. Suppose we are finding $0.023 - 0.007$. Here is a diagram showing 0.023.

We want to remove 7 thousandths (7 small rectangles). We can decompose one of the hundredths into 10 thousandths.

Base 10 diagram. 0 point 0 2 3. Two small squares in the hundredths column. Three small rectangles in the thousandths column. A square is drawn around 1 small square. An arrow is drawn to 10 small rectangles. The arrow is labeled decompose.

Now we can remove 7 thousandths.

Base 10 diagram. 0 point 0 2 3. One small square in the hundredths column. 13 small rectangles in the thousandths column. 7 small rectangles have an X through them. Below the small rectangles are the words subtract 0 point 0 0 7.

We have 1 hundredth and 6 thousandths remaining, so $0.023 - 0.007 = 0.016$.

Here is a vertical calculation of $0.023 - 0.007$.

Vertical subtraction. First line. 0 point 0 2 3. The 2 is crossed out and has a 1 above it. The 3 is crossed out and has 13 above it. Second line. Minus 0 point 0 0 7. Horizontal line. Third line. 0 point 0 1 6.

Base-ten diagrams work best for representing subtraction of numbers with few non-zero digits, such as $0.16 - 0.09$. For numbers with many non-zero digits, such as $0.25103 - 0.04671$, it would take a long time to draw the base-ten diagram. With vertical calculations, we can find this difference efficiently.

Thinking about base-ten diagrams can help us make sense of this calculation.

A setup for the subtraction calculation 0 point 2 5 1 0 3 subtract 0 point 0 4 6 7 1 results in 0 point 1 0 4 3 2. The number 0 point 2 5 1 0 3 is on top with the subtract 0 point 0 4 6 7 1 beneath, and the 0 from the first number lines up vertically with the 0 from the second number, the 2 from the first number lines up vertically with the 0 from the second, the 5 from the first number lines up vertically with the 4 from the second, and so on. The 1 in the thousandths place of the first number is unbundled to make ten groups of ten thousandths. The five in the hundredths place has 1 unbundled to make 4 hundredths and 10 thousandths.

The thousandth in 0.25103 is decomposed to make 10 ten-thousandths so that we can subtract 7 ten-thousandths. Similarly, one of the hundredths in 0.25103 is decomposed to make 10 thousandths.

We can use fractions like $\frac{1}{10}$ and $\frac{1}{100}$ to reason about the location of the decimal point in a product of two decimals.  

Let’s take $24 \boldcdot (0.1)$ as an example. There are several ways to find the product:

• We can interpret it as 24 groups of 1 tenth (or 24 tenths), which is 2.4.
• We can think of it as $24 \boldcdot \frac{1}{10}$, which is equal to $\frac {24}{10}$ (and also equal to 2.4).
• Because multiplying by $\frac {1}{10}$ has the same result as dividing by 10, we can also think of it as $24 \div 10$, which is equal to 2.4.

Here is another example: To find $(1.5) \boldcdot (0.43)$, we can think of 1.5 as 15 tenths and 0.43 as 43 hundredths. We can write the tenths and hundredths as fractions and rearrange the factors.

$\displaystyle \left(15 \boldcdot \frac{1}{10}\right) \boldcdot \left(43 \boldcdot \frac{1}{100}\right) = 15 \boldcdot 43 \boldcdot \frac{1}{1,000}$

Multiplying 15 and 43 gives us 645, and multiplying $\frac{1}{10}$ and $\frac{1}{100}$ gives us $\frac{1}{1,000}$. So $(1.5) \boldcdot (0.43)$ is $645 \boldcdot \frac{1}{1,000}$, which is 0.645.

To multiply two decimals, such as $(1.25) \boldcdot (0.7)$, we can multiply the whole numbers that have the same digits, $125 \boldcdot 7$, and then use what we know about place value to place the decimal point.

• Multiplying 125 and 7 gives 875.
• We know that 125 is 100 times 1.25, and 7 is 10 times 0.7, so the product of 125 and 7 is 1,000 times the product of 1.25 and 0.7.
• This means we need to divide 875 by 1,000, which moves the digits 3 places to the right and gives 0.875. 

Let’s find the product of 8.4 and 4.3!

• First, we multiply 84 and 43. 
• 84 is 10 times 8.4, and 43 is 10 times 4.3, so the product of 84 and 43 is 100 times the product of 8.4 and 4.3. 
• Dividing 3,612 by 100 moves the digits 2 places to the right, giving 36.12. 

Notice that:

• The factor 1.25 has 2 decimal places, the factor 0.7 has 1 decimal place, and the product 0.875 has 3 decimal places.
• The factors 8.4 and 4.3 each have 1 decimal place, and the product 36.12 has 2 decimal places.

In general, to find the product of decimals, we can first multiply the corresponding whole numbers. Then we can place the decimal point so the product has as many decimal places as the sum of decimal places in the factors.

Another way to find the quotient of $345 \div 3$ is by using partial quotients, in which we keep subtracting 3 groups of some amount from 345. We can organize the steps and record the partial quotients in a vertical calculation.

Here are two calculations for finding $345 \div 3$:

2 partial quotients methods of 345 divided by 3. First calculation, 11 rows. First row: 115. Second row: 5. Third row: 10. Fourth row: 100. Fifth row: 3, long division symbol with 345 inside. Sixth row: minus 300, 3 groups of 100. Horizontal line. Seventh row: 45. Eighth row: minus 30, 3 groups of 10. Horizontal line. Ninth row: 15. Tenth row: minus 15, 3 groups of 5. Horizontal line. Eleventh row: 0. Second calculation, 11 rows. First row: 115. Second row: 50. Third row: 50. Fourth row: 15. Fifth row: 3, long division symbol with 345 inside. Sixth row: minus 45, 3 groups of 15. Horizontal line. Seventh row: 300. Eighth row: minus 150, 3 groups of 50. Horizontal line. Ninth row: 150. Tenth row: minus 150, 3 groups of 50. Horizontal line. Eleventh row: 0

• In the calculation on the left, first we subtract 3 groups of 100, then 3 groups of 10, and then 3 groups of 5. Adding up the partial quotients ($100+10+5$) gives us 115.
• The calculation on the right shows a different amount per group subtracted each time (3 groups of 15, 3 groups of 50, and 3 more groups of 50), but the total amount in each of the 3 groups is still 115.

There are other ways of calculating $345 \div 3$ using partial quotients. We can calculate with fewer steps by removing groups of larger sizes.

We can use long division to find quotients even when the numbers involved are not whole numbers. Here is the long-division calculation of $86 \div 4$, which results in a decimal quotient.

Long division calculation of 86 divided by 4. 8 rows. First row: 21 point 5, Second row: 4, long division symbol with 86 inside. Third row: minus 8. Horizontal line. Fourth row: 6. Fifth row: minus 4. Horizontal line. Sixth row: 2 point 0. Seventh row: minus 2 point 0. Horizontal line. Eighth row: 0. 

The calculation shows that, after removing 4 groups of 21, there are 2 ones remaining. We can continue dividing by writing a 0 to the right of the 2 and thinking of that remainder as 20 tenths, which can then be divided into 4 groups.

To show that the quotient we are working with now is in the tenths place, we put a decimal point to the right of the 1 (which is in the ones place) at the top. It may also be helpful to draw a vertical line to separate the ones and the tenths.

There are 4 groups of 5 tenths in 20 tenths, so we write 5 in the tenths place at the top. The calculation likewise shows $86 \div 4 = 21.5$.

We know that two fractions are equivalent when the numerators and denominators are related by the same factor, and when dividing the numerator by the denominator gives the same quotient. For example, we can tell that $\frac{6}{4}$ and $\frac{60}{40}$ are equivalent fractions because:

• Dividing 6 by 4 and dividing 60 by 40 both give 1.5.
• The numerators and denominators of $\frac{6}{4}$ and $\frac{60}{40}$ are related by the same factor of 10: $\frac{6 \boldcdot 10}{4 \boldcdot 10} = \frac{60}{40}$.

Division expressions can also be equivalent. For example, the expression $5,400 \div 900$ is equivalent to $54 \div 9$ because:

• They both have a quotient of 6.
• The dividends and divisors in $5,400 \div 900$ and $54 \div 9$ are related by the same factor of 100: $54 \boldcdot 100 = 5,400$ and $9 \boldcdot 100 = 900$.

This means that an expression such as $5.4 \div 0.9$ also has the same value as $54 \div 9$. The dividend and divisor in $54 \div 9$ are each 10 times those in $5.4 \div 0.9$, but their quotients are the same.

This understanding can help us divide a decimal dividend by a decimal divisor: We can multiply each decimal by the same power of 10 so that both the dividend and the divisor are whole numbers, and then we divide the whole numbers.

For example, to calculate $7.65 \div 1.2$ we can multiply each decimal by 100, and then calculate $765 \div 120$. Here is the calculation with long division:

Long division of 765 divided by 120. 10 rows. Top row: 6 point 3 7 5. Second row: 120, long division symbol with 765 inside. Third row: minus 720. Horizontal line. Fourth row: 45 point 0. Fifth row: minus 36 point 0. Horizontal line. Sixth row: 9 point 0 0. Seventh row: minus 8 point 4 0. Horizontal line. Eighth row: point 6 0 0. Ninth row: minus point 6 0 0. Horizontal line. Tenth row: 0.

Because the expression $765 \div 120$ is equivalent to $7.65 \div 1.2$, we know that 6.375 is also a quotient of $7.65 \div 1.2$.

1. Find the sum: $3.7 + 14.56$

2. Find the difference: $8.5 - 3.27$

3. Find the difference: $3 - 1.5064$

4. Find the sum: $24.6 + 8.075$

5. Find the difference: $12.04 - 7.368$

6. A ribbon is 6.384 meters long. It is cut into 4 equal pieces. How long is each piece?

7. Find the product: $(0.24)(0.7)$

8. Find the product: $(0.035)(0.52)$

9. A baker used these amounts of flour for four recipes: 2.75 cups, 1.8 cups, 3.125 cups, and 4 cups.
(a) How much flour was used in total?
(b) How much more flour was used in the fourth recipe than the first?

10. A box of supplies costs $18.00. Each supply kit inside costs $7.50. How many kits are in the box?

1. A poster is 11.5 inches long and 8 inches wide. What is the area of the poster?

2. To find 10% of a number, you can multiply by 0.10. Find 10% of $85.00.

3. A store sells 6 water bottles for $15.00. What is the price per bottle?

4. A triangle has a base of 6.4 cm and a height of 5 cm. Find the area.
(Area of triangle $= \frac{1}{2} \times \text{base} \times \text{height}$)

5. To calculate a 20% tip on a meal, you can multiply the bill by 0.20. A restaurant bill is $36.75. What is the tip? Explain or show your reasoning.

6. Two stores sell the same notebook.
• Store A sells 4 notebooks for $9.00
• Store B sells 5 notebooks for $10.50
Which store has the lower price per notebook? Explain how you know.

7. A rectangular garden is 8.5 meters long and 4.2 meters wide. A triangular flower bed with a base of 4.2 meters and a height of 3 meters is cut out of one corner. What is the area of the remaining garden? Show your work.
(Area of triangle $= \frac{1}{2} \times \text{base} \times \text{height}$)

8. Maya buys 3 notebooks at $4.75 each and a pack of pens for $6.50. She pays with a $25.00 bill.
(a) How much did Maya spend in total?
(b) How much change did she receive?