P3: Decimals — Reading, Writing, and Operations

Goals: Convert between word and standard form of decimals, and perform addition, subtraction,
multiplication, and division with decimals (including sign rules and place-value alignment).

0. Decimal Place Value

From the decimal point moving right: tenths, hundredths, thousandths, ten-thousandths…

• [latex]0.7[/latex] = seven tenths
• [latex]0.78[/latex] = seventy-eight hundredths
• [latex]0.029[/latex] = twenty-nine thousandths

1. Writing Decimals from Words (and Vice Versa)

Rules to Go from Words → Number

1. Write the whole-number part (if none, write [latex]0[/latex]).
2. Say “and” only at the decimal point.
3. Write the fractional words as digits; the last word tells the place value.Example: “seventy eight hundredths” → [latex]0.78[/latex].

Rules to Go from Number → Words

1. Read the whole-number part.
2. Say “and” at the decimal point.
3. Read the digits to the right of the decimal as a whole number, then state the place value.Example: [latex]44.56[/latex] → “forty-four and fifty-six hundredths”.

2. Operations with Decimals

2.1 Adding & Subtracting Decimals

• Line up the decimal points in vertical form.
• Use placeholder zeros so each number has the same length (e.g., [latex]2.2 = 2.200[/latex]).
• Add/subtract as with whole numbers, then place the decimal in the aligned column.

2.2 Multiplying Decimals

1. Ignore decimal points and multiply as whole numbers.
2. Count total decimal places in the factors.
3. Place the decimal in the product so it has that many decimal places.
4. Apply sign rule: negative [latex]\times[/latex] positive is negative, etc.

2.3 Dividing Decimals

1. If the divisor is a decimal, move the decimal right until the divisor is a whole number, and move the dividend’s decimal the same number of places.
2. Divide as usual; bring the decimal straight up into the quotient.
3. Apply sign rule for division (same signs → positive; different signs → negative).

3. Practice Set

Writing Numbers from Words

1. seventy eight hundredths → [latex]0.78[/latex]
2. twenty nine thousandths → [latex]0.029[/latex]
3. thirty eight and sixty six hundredths → [latex]38.66[/latex]
4. four hundred sixty four and forty five hundredths → [latex]464.45[/latex]

Writing Words from Numbers

5. [latex]44.56[/latex] → forty-four and fifty-six hundredths
6. [latex]0.0018[/latex] → eighteen ten-thousandths
7. [latex]0.011[/latex] → eleven thousandths
8. two hundred one and forty-one hundredths → [latex]201.41[/latex]
9. forty-two thousandths → [latex]0.042[/latex]

Adding & Subtracting

10. [latex]\displaystyle 11.801 + 2.559 = 14.360[/latex]
11. [latex]\displaystyle 16.977 - 7.614 = 9.363[/latex]
12. [latex]\displaystyle 5.316 - 2.2 = 5.316 - 2.200 = 3.116[/latex]
13. [latex]\displaystyle 6.3 - 3.899 = 6.300 - 3.899 = 2.401[/latex]
14. [latex]\displaystyle 4.822 - 8.3 = 4.822 - 8.300 = -3.478[/latex]
15. [latex]\displaystyle 13.2 - 2.0 = 11.2[/latex]

Multiplying

16. [latex]\displaystyle 2.46 \times 6.52 = 16.0392[/latex]
17. [latex]\displaystyle 0.034 \times 0.0032 = 0.0001088[/latex]
18. [latex]\displaystyle 3.1 \times 3.41 = 10.571[/latex]
19. [latex]\displaystyle -6.25 \times 2.27 = -14.1875[/latex]
20. [latex]\displaystyle 0.088 \times (-0.0044) = -0.0003872[/latex]
21. [latex]\displaystyle -9.6 \times -1.31 = 12.576[/latex] (negative [latex]\times[/latex] negative = positive)

Dividing

22. [latex]\displaystyle 45.08 \div 9.2[/latex]:
    move decimals → [latex]\displaystyle 450.8 \div 92 = 4.9[/latex]
23. [latex]\displaystyle -14.95 \div 2.3[/latex]:
    move decimals → [latex]\displaystyle -149.5 \div 23 = -6.5[/latex]

Fraction & Decimal Connections

24. Evaluate in fractions; give a reduced fraction:[latex]\displaystyle \frac{7}{8} + 0.8 = \frac{7}{8} + \frac{8}{10} = \frac{7}{8} + \frac{4}{5} = \frac{35}{40} + \frac{32}{40} = \frac{67}{40}[/latex]
25. Multiply; give a decimal to the nearest hundredth:[latex]\displaystyle \left(\frac{7}{8}\right)\cdot 1.1 = \frac{7}{8}\cdot \frac{11}{10} = \frac{77}{80} = 0.9625 \approx 0.96[/latex]

4. Quick Checklist for Students

• Add/Subtract: Decimals aligned? Zeros added as placeholders?
• Multiply: Count total decimal places after multiplying the whole-number versions.
• Divide: Make the divisor a whole number by shifting both numbers the same number of places.
• Words/Numbers: Say “and” only at the decimal; use place value names correctly.
• Signs: Same signs → positive; different signs → negative.