MGU Mathematics Series — Arithmetic & Pre-Algebra Foundations
Part II: Fractions, Mixed Numbers, and Decimals
Place Value · Operations · Rounding · Fractions · Applications
Chapter 5 of 14
MGU Mathematics Series  ·  Arithmetic & Pre-Algebra Foundations  ·  Library Textbook Edition

Chapter 5:
Decimals

Place value, reading and writing, comparison, all four operations, rounding, repeating decimals, fraction conversion, and applications in money, measurement, and data

■ Part II: Fractions, Mixed Numbers, and Decimals ■

■ 14 Sections  •  Chapter 5 of 14 ■

Follows Chapter 4: Mixed Numbers  ·  Prepares students for Chapter 6: Ratios, Proportions, and Percents

Chapter 5 — Front Matter

Purpose and Overview

Chapter 5 teaches decimals as a natural extension of the base-ten place value system students already know from whole numbers. Decimals are not a separate subject invented to make arithmetic harder. They are the same system, extended to the right of the ones place, so that numbers smaller than one can be written and compared using the same logic as numbers larger than one.

Students who struggle with decimals often struggle not because the ideas are too hard, but because they were never shown the connection. They were taught to "line up the decimal points" without understanding why. They were taught to "move the decimal" without understanding what that means. This chapter slows down and builds that understanding from the ground up.

By the end of this chapter, students should be able to read, write, compare, and compute with decimals, convert between fractions and decimals, use estimation and rounding, and apply decimal reasoning to money, measurement, and data. These skills directly support the work of Chapters 6 through 14 and are essential preparation for Algebra I, personal finance, science, and technical coursework.

Opening Question

A track athlete finishes a race in 11.47 seconds. A competitor finishes in 11.5 seconds. The athlete looks at the scoreboard and thinks she lost, because 5 is greater than 4. Her coach tells her she actually won.

Who is right, and why? This question sits at the heart of Chapter 5. To answer it, you need to understand what digits to the right of a decimal point actually mean, how to compare decimal values correctly, and why zeros matter in ways that are not always visible. Hold this question in mind as you work through the chapter. You will be able to answer it with confidence by the time you reach Section 5.4.

14Sections
12Objectives
12Key Terms
15+Worked Examples

Why This Chapter Matters

Decimals appear in nearly every area of modern life. A price tag reads $14.99. A weather report gives 1.25 inches of rainfall. A chemistry lab measures 0.005 grams of a compound. A data table reports an average score of 87.6. A carpenter cuts a board to 3.75 feet. A doctor prescribes a dosage of 0.25 milligrams. In each of these cases, a whole number is not precise enough. Decimals allow quantities to be expressed with as much precision as needed, using the same base-ten structure that makes whole-number arithmetic work.

Beyond practical life, decimals connect arithmetic to fractions, percents, scientific notation, algebra, and statistics. Every topic in the remaining chapters of this book uses decimal notation in some way. Mastering decimals here means the later chapters are easier and faster.

Learning Objectives

  1. 1Identify the place value of any digit in a decimal number.
  2. 2Read and write decimals in standard form and in words.
  3. 3Compare and order decimals using place value reasoning.
  4. 4Plot decimals on a number line.
  5. 5Add and subtract decimals with and without regrouping.
  6. 6Multiply decimals and explain where to place the decimal point.
  7. 7Divide decimals, including cases with decimal divisors.
  8. 8Use estimation to check the reasonableness of decimal computations.
  9. 9Identify and write repeating decimals using bar notation.
  10. 10Round decimals to a specified place value.
  11. 11Convert between fractions and decimals in both directions.
  12. 12Apply decimal reasoning to money, measurement, and data problems.
Chapter 5 — Contents

Sections at a Glance

Sections 5.1 – 5.5
Place Value, Reading, Writing, and Comparing
Decimal Place Value · Tenths and Hundredths · Reading and Writing · Comparing · Number Lines
Section 5.1

Decimal Place Value

The base-ten number system is built on a simple idea: the value of any digit depends not only on what digit it is, but on where it sits in the number. This is the meaning of place value. In the number 3,472, the digit 3 does not mean three. It means three thousand, because it is in the thousands place. Each position to the left is worth ten times more than the position to its right.

Decimals extend this same pattern to the right of the ones place. Each position to the right of the decimal point is worth ten times less than the position to its left. The decimal point marks the boundary between the whole-number part and the fractional part. It is not a separator between two separate numbers. It is a marker that says: the whole-number part is on my left, and the fractional part is on my right.

The Extended Place Value Chart

Thousands Hundreds Tens Ones . Tenths Hundredths Thousandths Ten-Thousandths
1,000 100 10 1 · 1/10 1/100 1/1,000 1/10,000
← Whole-number part · Fractional part →
The "ths" Signal

The "ths" ending is the signal that a place is to the right of the decimal point and represents a fractional part. Tens and tenths are not the same. Hundreds and hundredths are not the same. This distinction matters enormously when reading and writing decimals.

Example 5.1.1
Identifying place values in a decimal number
  • ProblemIn the number 47.359, identify the place value of each digit.
  • ThinkRead the number from left to right. Identify which side of the decimal point each digit falls on, and count positions carefully.
  • Solve 4 → tens place → value: 40 7 → ones place → value: 7 3 → tenths place → value: 3/10 = 0.3 5 → hundredths place → value: 5/100 = 0.05 9 → thousandths place → value: 9/1000 = 0.009
  • CheckThe values should add to the original number: 40 + 7 + 0.3 + 0.05 + 0.009 = 47.359 ✓
Answer: Each digit holds a specific position; its value is the digit multiplied by that position's power of ten.
Student Warning

Do not confuse tens with tenths, or hundreds with hundredths. The "ths" ending always means a fractional position to the right of the decimal point. The number 0.06 is six hundredths; the number 60 is sixty. They are very different quantities.

Practice 5.1Decimal Place Value
  1. What place value is immediately to the right of the decimal point?
  2. How many times larger is the tenths place than the hundredths place?
  3. What is the place value of the digit 6 in 203.067?
  4. Write the value of each digit in 8.524 as a fraction.
  5. Write the value of each digit in 0.0031.
Section 5.2

Tenths, Hundredths, Thousandths, and Beyond

Tenths: The tenths place represents parts out of ten. The number 0.3 means three tenths, or 3/10. If you divide a whole into ten equal parts and shade three of them, you have 0.3.

Hundredths: The hundredths place represents parts out of one hundred. The number 0.07 means seven hundredths, or 7/100. Think of a ten-by-ten grid of 100 equal squares. Seven shaded squares represent 0.07.

Thousandths: The thousandths place represents parts out of one thousand. The number 0.004 means four thousandths, or 4/1000. The pattern continues beyond thousandths, and each position name ends in "ths."

NumberDigit 4 PositionFraction EquivalentRelationship
0.4Tenths4/10Base
0.04Hundredths4/100Ten times smaller than 0.4
0.004Thousandths4/1000Ten times smaller than 0.04
Zeros as Placeholders vs. Trailing Zeros

Placeholder zeros appear between the decimal point and a nonzero digit. In 0.003, the two zeros hold the tenths and hundredths places so the 3 lands in the thousandths place. Removing either zero changes the value.

Trailing zeros appear after the last nonzero digit. In 0.50, the trailing zero after 5 does not change the value: 0.5 = 0.50 = 0.500. They are sometimes used to show precision in measurement or to align columns in computation.

Student Tip — Zeros That Matter vs. Zeros That Don't

Trailing zeros after the decimal point do not change value. Zeros between the decimal point and a nonzero digit do change value because they are placeholders. The number 0.05 is not the same as 0.5.

Practice 5.2Tenths, Hundredths, Thousandths
  1. What fraction does 0.09 represent?
  2. Write 0.007 as a fraction.
  3. True or false: 0.60 and 0.6 are equal.
  4. Write as a fraction: (a) 0.1, (b) 0.01, (c) 0.001
  5. Which digit is in the hundredths place: 3.472?
  6. Explain why 0.3 and 0.03 are not equal.
Section 5.3

Reading and Writing Decimals

Reading a decimal number out loud uses a specific pattern. The decimal point is read as "and." To read a decimal: (1) read the whole-number part as a regular whole number; (2) say "and" for the decimal point; (3) read the digits to the right as a whole number; (4) say the name of the place value of the last digit on the right.

NumberHow to Read It
3.7Three and seven tenths
0.45Forty-five hundredths
12.008Twelve and eight thousandths
0.375Three hundred seventy-five thousandths
100.0001One hundred and one ten-thousandth
Student Warning — Placeholder Zeros When Writing

When writing a decimal from words, count the number of decimal places required by the place value name before writing any digits. "Sixteen thousandths" requires three decimal places. Sixteen is a two-digit number, so insert a placeholder zero: 0.016, not 0.16. Forgetting placeholder zeros is one of the most common errors in this section.

Example 5.3.1
Writing a decimal from words
  • ProblemWrite "two hundred and thirty-four thousandths" in decimal form.
  • ThinkWhole-number part: two hundred = 200. Fractional part: thirty-four thousandths. Thousandths requires three decimal places. Thirty-four is two digits, so I need a placeholder zero before it.
  • Solve200.034
  • CheckRead back: two hundred, and, zero-three-four in the thousandths position — thirty-four thousandths. ✓
Answer: 200.034
Example 5.3.2
Reading a decimal in words
  • ProblemWrite 5.0072 in words.
  • ThinkWhole-number part: 5. The decimal part is 0072. The last digit, 2, is in the ten-thousandths place. The four digits together represent 72.
  • SolveFive and seventy-two ten-thousandths
  • CheckCounting positions after the decimal: tenths (0), hundredths (0), thousandths (7), ten-thousandths (2). Four places, consistent with ten-thousandths. ✓
Answer: Five and seventy-two ten-thousandths.
Practice 5.3Reading and Writing Decimals

Write each decimal in words, then write each word form as a decimal.

  1. 0.8
  2. 3.14
  3. 0.065
  4. 102.0009
  5. Nine tenths
  6. Forty-seven hundredths
  7. Five and three thousandths
  8. One hundred twelve and sixteen ten-thousandths
Section 5.4

Comparing Decimals

Return now to the opening question. A runner finishes in 11.47 seconds. Her competitor finishes in 11.5 seconds. A common error is to compare 0.47 and 0.5 by looking at how many digits each has, and assuming that 47 is greater than 5, so 0.47 must be greater than 0.5. This reasoning is wrong because it ignores place value.

The correct approach is to compare digit by digit, from left to right, starting at the decimal point. To compare, line up the decimal points, add trailing zeros so both numbers have the same number of decimal places, then compare the digits from left to right. The first position where the digits differ tells you which number is larger.

Comparing Two Decimals — Step by Step

Step 1: Line up the decimal points.

Step 2: Add trailing zeros so both numbers have equal decimal places. (This does not change either value.)

Step 3: Compare digits from left to right, one position at a time.

Step 4: The first differing digit determines which number is greater.

Example 5.4.1
Resolving the opening question — comparing 11.47 and 11.5
  • ProblemWhich is greater: 11.47 or 11.5?
  • ThinkWrite 11.5 as 11.50 by adding a trailing zero. Now compare digit by digit.
  • Solve 11.47 11.50 Compare: tens=1=1, ones=1=1, tenths=4 vs 5 → 4 < 5
  • CheckAs fractions: 11.47 = 1147/100 and 11.50 = 1150/100. Since 1147 < 1150, we confirm 11.47 < 11.50. ✓
Answer: 11.47 < 11.50. The first runner's time of 11.47 seconds is smaller (faster). She won.
Example 5.4.2
Ordering three decimals
  • ProblemWrite 0.6, 0.062, and 0.60 in order from least to greatest.
  • ThinkAdd trailing zeros to make all three numbers have three decimal places: 0.600, 0.062, 0.600. Note that 0.6 and 0.60 are already equal.
  • Solve Compare 0.062 with 0.600: Tenths: 0 vs. 6 → 0 < 6 Order: 0.062 < 0.6 = 0.60
  • Check62/1000 = 0.062 and 600/1000 = 0.600. Since 62 < 600, the ordering is confirmed. ✓
Answer: 0.062 < 0.6 = 0.60
Student Warning — The Comparison Trap

Never compare decimals by counting digits. The number 0.47 has two decimal digits and 0.5 has one, but 0.5 is larger. Always compare place value by place value, lining up the decimal points first. The trap that misleads students is treating the digits after the decimal point as if they form a separate whole number. They do not.

Practice 5.4Comparing and Ordering Decimals

Use <, >, or = to compare. Then order each group from least to greatest.

  1. 0.3 __ 0.30
  2. 0.52 __ 0.6
  3. 1.08 __ 1.8
  4. 0.099 __ 0.1
  5. Order: 0.4, 0.04, 0.44, 0.404
  6. Order: 2.3, 2.03, 2.303, 2.033
  7. Is 0.9 greater than or less than 0.19? Explain.
  8. Error Analysis: A student says 0.75 > 0.8 because 75 > 8. What mistake did the student make?
Section 5.5

Decimal Number Lines

A number line is one of the most useful tools for understanding the relative size of decimal numbers. Decimals can be placed on a number line just as whole numbers and fractions can, because they represent specific points between whole-number values. To place a decimal on a number line, identify the two whole numbers it falls between, then divide that unit segment into the appropriate number of equal parts based on the decimal's precision.

One important property that number lines reveal about decimals is density. Between any two decimals, there is always another decimal. Between 0.1 and 0.2, there is 0.15. Between 0.1 and 0.15, there is 0.12. This can go on without limit. This is very different from whole numbers, where there is no whole number between 3 and 4. Decimal numbers fill in the gaps between whole numbers, making the number line continuous.

Decimals can also be negative. The number −0.5 is halfway between −1 and 0. A number further to the left on the number line is smaller: −1.2 < −0.8, even though 1.2 > 0.8 when both are positive.

Example 5.5.1
Locating decimals on a number line
  • ProblemDescribe where to place 3.6 and 3.06 on a number line from 3 to 4.
  • ThinkBoth numbers fall between 3 and 4. Divide the segment from 3 to 4 into ten equal tenths-segments.
  • Solve 3.6 → sixth tenths mark; six tenths of the way from 3 to 4 3.06 → tenths digit is 0; very close to 3, only six hundredths past it
  • Check3.6 = 3 + 0.6 and 3.06 = 3 + 0.06. Since 0.6 > 0.06, the point 3.6 is indeed farther from 3. ✓
Answer: 3.6 is at the sixth tenths mark. 3.06 is between the zero and first tenths marks, very close to 3.
Practice 5.5Decimal Number Lines
  1. Between which two whole numbers does 4.73 fall?
  2. Is −0.3 greater than or less than 0?
  3. On a number line from 0 to 1 divided into tenths, describe the approximate positions of 0.2, 0.5, 0.8, and 0.95.
  4. Which decimal is closer to 2 on a number line: 2.4 or 2.7?
  5. Place −1.5 and −1.05 on a number line from −2 to 0. Which is farther from 0?
  6. Application: A thermometer shows a reading between 98°F and 99°F, at the seventh tenth-mark. What temperature does it show?
Sections 5.6 – 5.10
Operations: Adding, Subtracting, Multiplying, Dividing, and Estimating
Adding · Subtracting · Multiplying · Dividing · Estimation
Section 5.6

Adding Decimals

Adding decimals uses the same algorithm as adding whole numbers, with one essential requirement: the decimal points must be aligned vertically before adding. When decimal points are aligned, every digit is automatically in its correct place value column. Without alignment, digits from different place values would be added together, producing a wrong answer.

Adding Decimals — The Core Procedure

Step 1: Line up the decimal points vertically.

Step 2: Add trailing zeros as needed so each number has the same number of decimal places.

Step 3: Add column by column from right to left, regrouping (carrying) when a column sums to ten or more.

Step 4: Place the decimal point in the answer directly below the aligned decimal points.

Student Tip — Write the Decimal Point First

Write the decimal point in your answer before you start adding. Placing it first, directly below the aligned decimal points, prevents errors in the final answer.

Example 5.6.1
Adding mixed whole and decimal numbers
  • ProblemAdd 12 + 3.7 + 0.045.
  • Think12 is a whole number. Treat it as 12.000 so it lines up with the others.
  • Solve 12.000 3.700 + 0.045 -------- 15.745
  • CheckEstimate: 12 + 4 + 0 ≈ 16. The answer 15.745 is close to 16. ✓
Answer: 15.745
Practice 5.6Adding Decimals
  1. 4.3 + 2.1
  2. 7.56 + 3.4
  3. 0.008 + 0.12
  4. 15 + 0.73 + 4.006
  5. 9.99 + 0.01
  6. 3.5 + 2.15 (show alignment)
Section 5.7

Subtracting Decimals

Subtracting decimals follows the same alignment rule as addition. Line up the decimal points, add trailing zeros as needed, and subtract column by column from right to left, regrouping (borrowing) when necessary. When subtracting a decimal from a whole number, write the whole number with a decimal point and enough trailing zeros to match the decimal being subtracted.

Example 5.7.1
Subtracting with regrouping
  • ProblemCompute 6.3 − 2.87.
  • ThinkWrite 6.3 as 6.30. Borrow as needed, working right to left.
  • Solve 6.30 - 2.87 ------ 3.43
  • Check3.43 + 2.87 = 6.30 ✓
Answer: 3.43
Example 5.7.2
Subtracting a decimal from a whole number
  • ProblemA rope is 5 meters long. A piece 1.364 meters long is cut off. How much rope remains?
  • ThinkCompute 5 − 1.364. Write 5 as 5.000.
  • Solve 5.000 - 1.364 ------- 3.636
  • Check3.636 + 1.364 = 5.000 ✓
Answer: 3.636 meters of rope remain.
Student Warning — Borrowing Step by Step

When borrowing in subtraction, work one column at a time. It is easy to make errors if you try to do multiple borrows in your head at once. Write each step clearly. When subtracting from a whole number, always rewrite it with a decimal point and trailing zeros before beginning.

Practice 5.7Subtracting Decimals
  1. 9.5 − 3.2
  2. 7.04 − 2.63
  3. 5 − 1.87
  4. 10 − 0.009
  5. 14.3 − 9.475
  6. You have $20.00 and spend $13.47. How much is left?
Section 5.8

Multiplying Decimals

When multiplying decimals, you do not need to align decimal points before computing. Instead, multiply the numbers as if they were whole numbers and then count decimal places to determine where the decimal point belongs in the product. This works because of how fractions with powers-of-ten denominators multiply. When you multiply tenths by tenths, you get hundredths:

Multiplying Decimals — The Procedure

Step 1: Ignore the decimal points and multiply as if both numbers were whole numbers.

Step 2: Count the total number of decimal places in both factors (digits to the right of each decimal point).

Step 3: Place the decimal point in the product so that it has exactly that total number of decimal places, counting from the right.

Student Tip — Products Smaller Than Their Factors

When multiplying two numbers that are each less than one, the product will be smaller than either factor. For example, 0.5 × 0.5 = 0.25. This surprises some students, but it makes sense because you are taking half of a half.

Example 5.8.1
Multiplying two decimals
  • ProblemMultiply 3.6 × 0.025.
  • ThinkMultiply 36 × 25, then count decimal places. 3.6 has one decimal place; 0.025 has three. Total: four decimal places.
  • Solve 36 × 25 = 900 Four decimal places from the right: 0.0900 Remove trailing zero: 0.09
  • CheckEstimate: 3.6 ≈ 4 and 0.025 ≈ 0.025; then 4 × 0.025 = 0.1. Our answer 0.09 is close. ✓
Answer: 3.6 × 0.025 = 0.09
Example 5.8.2
Real-world multiplication
  • ProblemA bag of rice weighs 2.75 pounds. You buy 6 bags. How much rice is that?
  • Solve 6 × 275 = 1650 2.75 has two decimal places; 6 has zero. Total: two decimal places → 16.50
  • CheckEstimate: 6 × 3 = 18. Our answer 16.50 is close. ✓
Answer: 16.5 pounds.
Practice 5.8Multiplying Decimals
  1. 0.3 × 0.9
  2. 4.5 × 3
  3. 1.2 × 0.04
  4. 7.3 × 2.1
  5. 0.006 × 0.5
  6. 0.08 × 10,000
  7. A car travels 32.5 mi/hr for 2.5 hours. How far?
  8. Each bottle: 1.75 L. A crate has 12. Total volume?
Section 5.9

Dividing Decimals

The simplest case of decimal division is dividing a decimal by a whole number: perform the division as you would with whole numbers and place the decimal point in the quotient directly above its position in the dividend. When the divisor is a decimal, eliminate the decimal in the divisor before dividing by multiplying both the dividend and the divisor by the same power of ten. This works because dividing both numbers by the same amount does not change the quotient.

Dividing by a Decimal — Eliminating the Decimal Divisor

Step 1: Count the number of decimal places in the divisor.

Step 2: Multiply both the dividend and divisor by 10 raised to that number.

Step 3: The divisor is now a whole number. Divide as usual and place the decimal point carefully.

Example 5.9.1
Dividing by a decimal
  • ProblemCompute 3.15 ÷ 0.05.
  • ThinkThe divisor 0.05 has two decimal places. Multiply both numbers by 100.
  • Solve 3.15 × 100 = 315 0.05 × 100 = 5 315 ÷ 5 = 63
  • Check63 × 0.05 = 3.15 ✓
Answer: 3.15 ÷ 0.05 = 63
Example 5.9.2
Measurement application
  • ProblemA board is 7.2 inches long. Each piece must be 0.6 inches. How many pieces can be cut?
  • Solve 7.2 ÷ 0.6 Multiply both by 10: 72 ÷ 6 = 12
  • Check12 × 0.6 = 7.2 ✓
Answer: 12 pieces.
Student Warning — Multiply Both Numbers

When converting a decimal-divisor problem, multiply the dividend and divisor by the same power of ten. If you only move the decimal in the divisor without making the corresponding change to the dividend, the answer will be wrong.

Practice 5.9Dividing Decimals
  1. 4.8 ÷ 6
  2. 0.72 ÷ 0.9
  3. 15 ÷ 0.03
  4. 2.856 ÷ 0.4
  5. 9 ÷ 0.005
  6. 0.48 ÷ 0.06
  7. A recipe needs 0.25 cups/serving. 2 cups total → how many servings?
  8. Stack of books: 7.2 inches tall, each 0.6 inches thick. How many books?
Section 5.10

Decimal Arithmetic and Estimation

Estimation is not the same as guessing. It is a deliberate, reasoned process of finding an approximate answer before or after computing, to determine whether an answer is in the right range. In decimal arithmetic, estimation is especially valuable because decimal point placement errors are common and often hard to catch without a reference point. A student who estimates first will immediately notice if a computed answer is ten times too large or ten times too small.

Estimation Strategies

Front-end rounding: Round each number to the largest place value it contains, then compute with the rounded values. Example: 8.73 + 3.12 → round to 9 + 3 = 12. Actual: 11.85.

Compatible numbers: Look for pairs or groups that are easy to compute with mentally. Example: 19.8 ÷ 3.9 → treat as 20 ÷ 4 = 5. Actual: approximately 5.08.

Student Tip — Always Estimate Before Multiplying or Dividing

Always estimate before or after a decimal multiplication or division. If your estimate and your computed answer differ by a factor of ten or more, you have almost certainly placed the decimal point in the wrong position.

Example 5.10.1
Estimation in a shopping situation
  • ProblemYou buy 4 items priced at $3.98, $5.12, $2.89, and $4.47. About how much do you spend?
  • ThinkRound each price to the nearest dollar, then add.
  • Solve $4 + $5 + $3 + $4 = $16
  • CheckExact sum: $3.98 + $5.12 + $2.89 + $4.47 = $16.46. Our estimate of $16 is very close. ✓
Answer: About $16. The exact total is $16.46.
Practice 5.10Estimation

Estimate each result first, then compute exactly.

  1. 6.9 + 4.3
  2. 8.05 × 3.9
  3. 23.6 ÷ 5.8
  4. 14.95 − 6.13
  5. Quiz scores: 7.5, 8, 6.9, 9.1, 8.4. Estimate the sum, then find the exact sum.
  6. A bottle of juice costs $2.79. About how much for 8 bottles?
Sections 5.11 – 5.14
Repeating Decimals, Rounding, Fractions, and Applications
Repeating Decimals · Rounding · Fraction–Decimal Conversion · Money, Measurement, and Data
Section 5.11

Repeating Decimals

Until now, all of the decimals in this chapter have ended after a certain number of digits. These are called terminating decimals. But not all decimals end. Some decimals go on forever in a repeating pattern. These are called repeating decimals, and they arise when a fraction is converted to decimal form by division and the remainder never reaches zero.

The most familiar example is the fraction 1/3. When you divide 1 by 3, you get 0.333333…, the digit 3 repeating without ever stopping. If you perform long division and the same remainder appears twice, the decimal pattern will repeat from that point on.

Bar Notation for Repeating Decimals

Writing repeating decimals as a long string of digits is impractical. Instead, a horizontal bar is written over the digit or digits that repeat.

0.333… = 0.‾3

0.142857142857… = 0.‾142857

0.727272… = 0.‾72

0.16666… = 0.1‾6 (only the 6 repeats; the 1 does not)

Example 5.11.1
Finding a repeating decimal by long division
  • ProblemPerform long division to find the decimal form of 5/6. Write using bar notation.
  • ThinkDivide 5 by 6 and track remainders. Watch for a repeating remainder.
  • Solve 5 ÷ 6 = 0 remainder 5 → 50 50 ÷ 6 = 8 remainder 2 → digit: 8 20 ÷ 6 = 3 remainder 2 → digit: 3 20 ÷ 6 = 3 remainder 2 → repeats Result: 0.8333… = 0.8‾3
  • CheckThe remainder 2 keeps producing 3s. The pattern is confirmed. ✓
Answer: 5/6 = 0.8‾3
Terminating vs. Repeating — A Preview

Every fraction in lowest terms produces a terminating decimal if and only if the denominator's only prime factors are 2 and 5. Any other prime factor in the denominator will cause the decimal to repeat. This connects to the prime factorization ideas from Chapter 2 and the fraction concepts from Chapter 3.

MGU Library Connection
  • The Fraction-Decimal Conversion Chart in Appendix D lists common fractions with their decimal equivalents, including which are terminating and which are repeating.
  • The connection between prime factors of denominators and decimal type is developed further in Chapter 7 (Exponents and Radicals) and Chapter 2 (Factors and Multiples).
Practice 5.11Repeating Decimals

Write each fraction as a decimal. Use bar notation if the decimal repeats.

  1. 1/3
  2. 5/9
  3. 7/11
  4. 3/8
  5. 2/3
  6. 1/6
  7. Error Analysis: A student writes 0.333 = 0.‾333 as the decimal for 1/3. What is wrong with this notation?
  8. What is the difference between a terminating and a repeating decimal?
Section 5.12

Rounding Decimals

Rounding replaces a number with a nearby, simpler value that is easier to work with or report. Rounding does introduce approximation: the rounded value is not exactly equal to the original. The goal is to lose as little accuracy as possible while gaining simplicity.

The Rounding Rule

Step 1: Identify the target rounding place.

Step 2: Look at the digit immediately to the right of the target place (the "rounding digit").

Step 3: If the rounding digit is 5 or greater, round up: increase the target digit by 1.

Step 4: If the rounding digit is less than 5, round down: leave the target digit unchanged.

Step 5: Drop all digits to the right of the target place.

NumberTarget PlaceRounding DigitResult
3.748Tenths4 (hundredths) < 5 → round down3.7
3.748Hundredths8 (thousandths) ≥ 5 → round up3.75
3.748Whole number7 (tenths) ≥ 5 → round up4
4.997Hundredths7 ≥ 5 → round up → carry through nines5.00
Student Warning — Look Only One Place Right

When rounding, look only at the one digit immediately to the right of the target place. Do not try to "chain" the rounding through multiple digits. Rounding 3.745 to the nearest tenth does not require you to first round to hundredths. Look only at the digit in the hundredths place, which is 4, and round down.

Example 5.12.1
Rounding in a financial context
  • ProblemA store divides a cost of $14.5575 equally among 4 customers. Each owes $3.638... Round to the nearest cent.
  • ThinkRound 3.638... to the nearest hundredth. Hundredths digit: 3. Rounding digit: thousandths → 8. Since 8 ≥ 5, round up.
  • Solve Hundredths digit 3 + 1 = 4 → $3.64
  • Check4 × $3.64 = $14.56. Differs from $14.5575 by a tiny rounding difference. Expected and acceptable. ✓
Answer: Each customer owes $3.64.
Practice 5.12Rounding Decimals

Round each number to the indicated place.

  1. 6.732 → nearest tenth
  2. 0.0856 → nearest thousandth
  3. 12.499 → nearest whole number
  4. 3.1415926 → nearest hundredth
  5. 0.995 → nearest hundredth
  6. 8.999 → nearest tenth
  7. Runner's time: 13.5284 seconds. Round to nearest hundredth for the scoreboard.
  8. Fuel price: $3.499/gallon. Round to nearest cent.
Section 5.13

Fractions and Decimals

Fractions and decimals are two different ways of writing the same kinds of numbers. Any fraction can be written as a decimal, and any terminating or repeating decimal can be written as a fraction. Both are representations of the same quantity; neither is more "real" than the other. Students who are comfortable moving between fractions and decimals have much greater flexibility in computation and problem-solving.

Fraction → Decimal: Divide the Numerator by the Denominator

Converting a Fraction to a Decimal

Divide: numerator ÷ denominator

3/4 = 3 ÷ 4 = 0.75 (terminates)

1/3 = 1 ÷ 3 = 0.‾3 (repeats)

7/8 = 7 ÷ 8 = 0.875 (terminates)

Decimal → Fraction: Use a Power-of-Ten Denominator

Converting a Terminating Decimal to a Fraction

Step 1: Write the decimal digits as the numerator.

Step 2: Write a power of ten as the denominator, matching the number of decimal places.

Step 3: Simplify the fraction.

0.3 = 3/10

0.75 = 75/100 = 3/4

0.125 = 125/1000 = 1/8

Repeating Decimal → Fraction: Algebraic Method

Converting a repeating decimal to a fraction uses a brief algebraic technique. The key idea is to set up an equation, then subtract to eliminate the repeating part.

Example 5.13.1
Converting a repeating decimal to a fraction
  • ProblemConvert 0.‾4 to a fraction.
  • ThinkLet x = 0.4444…. Multiply both sides by 10 to shift the decimal one place, then subtract.
  • Solve Let x = 0.4444… 10x = 4.4444… 10x − x = 4.4444… − 0.4444… 9x = 4 x = 4/9
  • Check4 ÷ 9 = 0.‾4 ✓
Answer: 0.‾4 = 4/9
Example 5.13.2
Full conversion — fraction to decimal
  • ProblemConvert 9/16 to decimal form.
  • ThinkDivide 9 by 16 using long division. The denominator 16 = 2⁴, so the decimal will terminate.
  • Solve 90 ÷ 16 = 5 remainder 10 100 ÷ 16 = 6 remainder 4 40 ÷ 16 = 2 remainder 8 80 ÷ 16 = 5 exactly Result: 0.5625
  • Check0.5625 × 16 = 9 ✓
Answer: 9/16 = 0.5625
MGU Library Connection
  • The Fraction-Decimal Conversion Chart in Appendix D includes the most common fraction-decimal equivalents.
  • Memorizing key equivalents such as 1/4 = 0.25, 1/3 = 0.‾3, and 1/8 = 0.125 will save significant time in Chapters 6–14.
Practice 5.13Fractions and Decimals

Convert each fraction to a decimal. Then convert each decimal to a simplified fraction.

  1. 1/2
  2. 3/5
  3. 7/9
  4. 5/16
  5. 11/12
  6. 4/11
  7. 0.4 → fraction
  8. 0.36 → fraction
  9. 0.008 → fraction
  10. 0.625 → fraction
  11. Challenge: Use the algebraic method to convert 0.‾27 to a fraction.
Section 5.14

Decimals in Money, Measurement, and Data

In real-world contexts, decimals appear in three major areas: financial calculations, measurement, and data reporting. Each context has its own conventions and requires careful attention to decimal precision.

Money: Currency in the United States uses exactly two decimal places — dollars and cents. Always round final monetary answers to the nearest cent. Do not round intermediate steps; round only the final answer.

Measurement: Scientific and technical measurements often require decimal precision. In the metric system, all units are based on powers of ten, which makes decimal arithmetic especially natural. When computing with measurements, always keep track of units. Adding 1.2 meters and 45 centimeters requires converting to the same unit first: 1.2 m + 0.45 m = 1.65 m.

Data: When data is collected, averages (means), medians, and rates are often expressed as decimals even when the original data consisted of whole numbers. An average of 81.6 is a perfectly valid and important number, even though no single measurement was 81.6.

Example 5.14.1
Budgeting with decimals
  • ReadEarnings: $52.50/week. Expenses: $18.75 food + $9.50 transportation + $7.00 entertainment.
  • RepresentSavings = Earnings − Total Expenses
  • Solve Total expenses: 18.75 + 9.50 + 7.00 = 35.25 Savings: 52.50 − 35.25 = 17.25
  • Check35.25 + 17.25 = 52.50 ✓
  • ExplainAfter paying for food, transportation, and entertainment, you save $17.25 each week.
Answer: $17.25 per week in savings.
Example 5.14.2
Measurement with decimals
  • ProblemA board is 2.4 meters long. It needs to be cut into pieces each 0.15 meters long. How many pieces?
  • Solve 2.4 ÷ 0.15 Multiply both by 100: 240 ÷ 15 = 16
  • Check16 × 0.15 = 2.40 ✓
Answer: 16 pieces.
Practice 5.14Real-World Applications
  1. You buy groceries for $43.78 and pay with $50.00. How much change should you receive?
  2. A recipe requires 0.75 liters of milk. You have 3.2 liters. After making the recipe, how much milk is left?
  3. Five friends run: 3.2, 4.75, 2.9, 5.0, and 3.85 km. What is the total distance? What is the average distance per person?
  4. A rope is 12 meters long. It is cut into pieces of 0.8 meters each. How many pieces result?
  5. A car travels 346.5 miles on 11.5 gallons of gas. What is the fuel efficiency in miles per gallon?
  6. Error Analysis: A student calculates that 6 notebooks at $2.35 each cost $1.41. Without computing the exact answer, explain how estimation shows this answer is wrong.
Chapter 5 — Back Matter

Chapter Summary

Key Terms Review

decimal
A number written using a decimal point to separate the whole-number part from the fractional part.
decimal point
The dot that separates the ones place from the tenths place. Read aloud as "and."
place value
The value of a digit based on its position in a number. Each position is ten times the value of the position to its right.
tenths
The first place to the right of the decimal point, representing parts out of ten. The number 0.3 means three tenths.
hundredths
The second place to the right of the decimal point, representing parts out of one hundred.
thousandths
The third place to the right of the decimal point, representing parts out of one thousand.
terminating decimal
A decimal that ends after a finite number of digits. Example: 0.75 terminates after two decimal places.
repeating decimal
A decimal in which one or more digits repeat in a pattern without end. Example: 0.333… = 0.‾3.
bar notation
A horizontal bar written over the repeating digit or digits in a repeating decimal, showing exactly which part repeats.
equivalent decimals
Decimals that represent the same value. Example: 0.5, 0.50, and 0.500 are all equal.
rounding
Replacing a number with a nearby value that is easier to work with, keeping a specified number of decimal places.
estimation
Finding an approximate answer before or after computing, to check whether a result is reasonable. Not the same as guessing.

Concept Review Questions

Answer in complete sentences.

  1. Explain why the base-ten place value system extends naturally to decimals.
  2. What is the role of a placeholder zero in a decimal number? Give an example where removing a zero would change the value.
  3. Explain why 0.5 and 0.50 are equal, but 0.5 and 0.05 are not.
  4. Why must decimal points be aligned before adding or subtracting, but not before multiplying?
  5. Describe the rule for placing the decimal point in a product. Why does this rule work?
  6. How do you eliminate a decimal from the divisor in a division problem? Give an example.
  7. Why is estimation valuable in decimal computation?
  8. What is the difference between a terminating decimal and a repeating decimal? What determines which a fraction will produce?
  9. Describe the rounding rule. What happens when the rounding digit is exactly 5?
  10. Why is it important to be precise about decimal places when working with money?

Place Value Review

  1. In the number 305.0274, identify the place value of every digit.
  2. Write the value of the digit 7 in each number: (a) 7.12, (b) 0.007, (c) 700.3, (d) 0.07
  3. Which number has a 5 in the hundredths place: 5.61, 0.05, 0.51, or 50.1?
  4. Write a decimal number where the digit 4 is in the thousandths place and the digit 9 is in the tenths place.

Decimal Computation Practice

Part AAddition
  1. 5.4 + 3.7
  2. 12.06 + 8.9
  3. 0.007 + 0.03 + 0.1
  4. 104 + 0.056 + 3.4
Part BSubtraction
  1. 9.1 − 4.8
  2. 20 − 14.37
  3. 6.003 − 2.1
  4. 100 − 0.01
Part CMultiplication
  1. 0.6 × 0.4
  2. 5.5 × 2.4
  3. 0.003 × 0.07
  4. 14.5 × 100
Part DDivision
  1. 8.1 ÷ 9
  2. 0.48 ÷ 0.06
  3. 3 ÷ 0.004
  4. 72 ÷ 0.8

Fraction–Decimal Conversion

Part AFractions to Decimals

State whether each decimal terminates or repeats.

  1. 1/4
  2. 2/3
  3. 5/8
  4. 4/11
  5. 7/20
  6. 5/6
Part BDecimals to Fractions in Simplest Form
  1. 0.25
  2. 0.64
  3. 0.005
  4. 0.375
Part CBar Notation

Write each repeating decimal using bar notation.

  1. 0.666666…
  2. 0.181818…
  3. 0.123123123…
  4. 0.58333… (only the 3 repeats)

Word Problems

Use the Read → Represent → Solve → Check → Explain method.

  1. A cell phone plan costs $35.99 per month. How much does the plan cost per year?
  2. A garden hose delivers water at 2.8 gallons per minute. How many gallons are delivered in 12.5 minutes?
  3. Your electricity bill is $87.46 this month and $72.80 last month. By how much did the bill increase?
  4. You have a container that holds 3.75 liters. You pour out 0.625 liters, then add 1.4 liters. How much is in the container now?
  5. A clothing store has 144 shirts. Each shirt costs $8.49 wholesale. What is the total wholesale cost of all the shirts?
  6. Three runners complete a relay race. Their times are 12.34 seconds, 11.9 seconds, and 13.05 seconds. What is their total time? What is their average time per runner?
  7. A scientist measures the mass of a compound as 0.0048 grams. She divides it equally into 8 samples. What is the mass of each sample?
  8. A car's odometer reads 42,510.7 miles at the start of a trip and 42,786.3 miles at the end. How long was the trip?

Error Analysis

Each problem shows a common student error. Identify the mistake, explain what went wrong, and give the correct answer.

Problem 1
0.12 > 0.9 because 12 > 9

What is the student's error? What is the correct comparison?

Problem 2
4.5 × 2.3 = 103.5

Use estimation to show why this is wrong. What is the correct answer?

Problem 3
1/3 = 0.‾333 (bar over all three digits)

What is wrong with this bar notation? Write the correct version.

Problem 4
When subtracting 6.48 ÷ 0.4, a student multiplies only the divisor by 10, getting 6.48 ÷ 4 = 1.62

What step did the student skip? What is the correct answer?

Problem 5
Round 3.745 to the nearest tenth: student looks at 4 and 5, rounds 4 up to 5, then rounds 7 up to 8 → answer: 3.8

What rounding error did the student make? What is the correct answer?

Problem 6
6 notebooks at $2.35 each cost $1.41

Without computing exactly, use estimation to show this answer is impossible. What likely went wrong?

Chapter 5 Checkpoint

Chapter 5 Checkpoint
Complete without assistance — assesses all key skills and concepts from Chapter 5
Part A — Place Value and Number Sense
  1. What is the place value of the digit 3 in 0.0304?
  2. Write 0.0082 in words.
  3. Write "four hundred and six thousandths" in decimal form.
  4. Order from least to greatest: 0.6, 0.59, 0.609, 0.061.
Part B — Computation
  1. 7.3 + 4.85 + 0.007
  2. 15 − 9.63
  3. 2.4 × 0.05
  4. 8.4 ÷ 0.7
  5. 0.009 × 1000
  6. 0.072 ÷ 0.009
Part C — Fractions and Decimals
  1. Convert 3/8 to a decimal.
  2. Convert 5/12 to a decimal using bar notation.
  3. Write 0.64 as a fraction in simplest form.
  4. Write 0.007 as a fraction.
Part D — Rounding and Estimation
  1. Round 6.3847 to the nearest hundredth.
  2. Round 0.0952 to the nearest thousandth.
  3. Estimate the product 8.9 × 5.1, then compute exactly.
Part E — Applications
  1. You buy two items for $12.75 and $8.49 and pay with $25.00. How much change do you get?
  2. A board 9.6 meters long is cut into pieces 0.4 meters each. How many pieces result?
  3. Your test scores are 88.5, 76, 91.4, 83, and 80.1. What is your average score?

Answers may be placed in the answer key.

Bridge Note — Moving Toward Chapter 6
Chapter 6: Ratios, Proportions, and Percents

Chapter 5 built your fluency with decimal notation and computation. In Chapter 6, you will encounter one of the most important applications of decimal reasoning: percents. A percent is simply a ratio expressed as parts per hundred, and it is computed and applied using decimals. A sales tax of 7.5% becomes the decimal 0.075. A discount of 20% means multiplying a price by 0.80. Understanding decimal multiplication and place value, which you practiced throughout this chapter, makes percent calculation a natural next step. Chapter 6 also introduces proportional reasoning, unit rates, and simple interest, all of which use decimal arithmetic in practical and meaningful ways. The decimal skills from Chapter 5 will appear in every remaining chapter of this textbook and in every MGU mathematics course that follows.

MGU Library Connection
  • Fraction, Decimal, and Percent Conversion Chart — Appendix D of this textbook
  • Place Value Reference Chart — Appendix B of this textbook
  • Chapter 2 (Factors and Multiples): prime factors of denominators determine whether a fraction terminates or repeats
  • Chapter 3 (Fractions): fraction-to-decimal conversion connects directly to division of fractions
  • Chapter 6 (Ratios, Proportions, and Percents): percents are decimals multiplied by 100
  • Chapter 8 (Scientific Notation): decimal place value and powers of ten underlie all scientific notation work
  • MGU Dictionary entries: decimal, place value, terminating decimal, repeating decimal, rounding, estimation