Question

to divide decimals, its important to get rid of the decimal point from the number we are dividing by.

• multiply the divisor by as many 10s as necessary to get a whole number.
• multiply the dividend by the same number of 10s.

example:
\\\

$$\begin{array}{r}22.1 \\\\ 3.5 \\overline{)177.35} \\\\ \\underline{-770.0} \\\\ 35 \\\\ \\underline{\\ \\ \\ 35} \\\\ 0 \\\\ \\end{array}$$

\\

a. \\\

$$\begin{array}{r}2.022 \\\\ 30 \\overline{)60.660} \\\\ \\underline{-60 \\ \\ \\ \\ \\ \\ } \\\\ 066 \\\\ \\underline{\\ \\ -60 \\ \\ \\ } \\\\ 60 \\\\ \\underline{\\ \\ -60 \\ \\ } \\\\ 0 \\\\ \\end{array}$$

\\ b. \\( 4.2 \overline{)20.076} \\) c. \\( 5.06 \overline{)47.058} \\) d. \\( 7 \overline{)24.36}
e. \\( .08 \overline{)3.7872} \\) f. \\( 8.4 \overline{)82.152} \\) g. \\( 3.5 \overline{)9.205} \\) h. \\( 6.02 \overline{)56.588}
i. \\( 10 \overline{)84.6} \\) j. \\( 7.2 \overline{)22.104} \\) k. \\( 5.67 \overline{)41.391} \\)

Answer:

Let's solve each division problem one by one:

Part a: \( 30 \enclose{longdiv}{60.660} \)

We can see from the given work that the quotient is \( 2.022 \). Let's verify: \( 30 \times 2.022 = 60.66 \), which matches the dividend. So the answer is \( 2.022 \).

Part b: \( 4.2 \enclose{longdiv}{20.076} \)

To divide by a decimal, multiply divisor and dividend by \( 10 \) to make divisor whole: \( 42 \enclose{longdiv}{200.76} \)

• \( 42 \times 4 = 168 \), \( 200.76 - 168 = 32.76 \)
• \( 42 \times 0.78 = 32.76 \) (since \( 42 \times 0.7 = 29.4 \), \( 42 \times 0.08 = 3.36 \); \( 29.4 + 3.36 = 32.76 \))
• So total quotient: \( 4 + 0.78 = 4.78 \)

Check: \( 4.2 \times 4.78 = 20.076 \). Correct. Answer: \( 4.78 \)

Part c: \( 5.06 \enclose{longdiv}{47.058} \)

Multiply divisor and dividend by \( 100 \): \( 506 \enclose{longdiv}{4705.8} \)

• \( 506 \times 9 = 4554 \), \( 4705.8 - 4554 = 151.8 \)
• \( 506 \times 0.3 = 151.8 \)
• Quotient: \( 9 + 0.3 = 9.3 \)

Check: \( 5.06 \times 9.3 = 47.058 \). Correct. Answer: \( 9.3 \)

Part d: \( 7 \enclose{longdiv}{24.36} \)

• \( 7 \times 3 = 21 \), \( 24.36 - 21 = 3.36 \)
• \( 7 \times 0.48 = 3.36 \) (since \( 7 \times 0.4 = 2.8 \), \( 7 \times 0.08 = 0.56 \); \( 2.8 + 0.56 = 3.36 \))
• Quotient: \( 3 + 0.48 = 3.48 \)

Check: \( 7 \times 3.48 = 24.36 \). Correct. Answer: \( 3.48 \)

Part e: \( 0.08 \enclose{longdiv}{3.7872} \)

Multiply divisor and dividend by \( 100 \): \( 8 \enclose{longdiv}{378.72} \)

• \( 8 \times 47 = 376 \), \( 378.72 - 376 = 2.72 \)
• \( 8 \times 0.34 = 2.72 \) ( \( 8 \times 0.3 = 2.4 \), \( 8 \times 0.04 = 0.32 \); \( 2.4 + 0.32 = 2.72 \))
• Quotient: \( 47 + 0.34 = 47.34 \)

Check: \( 0.08 \times 47.34 = 3.7872 \). Correct. Answer: \( 47.34 \)

Part f: \( 8.4 \enclose{longdiv}{82.152} \)

Multiply divisor and dividend by \( 10 \): \( 84 \enclose{longdiv}{821.52} \)

• \( 84 \times 9 = 756 \), \( 821.52 - 756 = 65.52 \)
• \( 84 \times 0.78 = 65.52 \) ( \( 84 \times 0.7 = 58.8 \), \( 84 \times 0.08 = 6.72 \); \( 58.8 + 6.72 = 65.52 \))
• Quotient: \( 9 + 0.78 = 9.78 \)

Check: \( 8.4 \times 9.78 = 82.152 \). Correct. Answer: \( 9.78 \)

Part g: \( 3.5 \enclose{longdiv}{9.205} \)

Multiply divisor and dividend by \( 10 \): \( 35 \enclose{longdiv}{92.05} \)

• \( 35 \times 2 = 70 \), \( 92.05 - 70 = 22.05 \)
• \( 35 \times 0.63 = 22.05 \) ( \( 35 \times 0.6 = 21 \), \( 35 \times 0.03 = 1.05 \); \( 21 + 1.05 = 22.05 \))
• Quotient: \( 2 + 0.63 = 2.63 \)

Check: \( 3.5 \times 2.63 = 9.205 \). Correct. Answer: \( 2.63 \)

Part h: \( 6.02 \enclose{longdiv}{56.588} \)

Multiply divisor and dividend by \( 100 \): \( 602 \enclose{longdiv}{5658.8} \)

• \( 602 \times 9 = 5418 \), \( 5658.8 - 5418 = 240.8 \)
• \( 602 \times 0.4 = 240.8 \)
• Quotient: \( 9 + 0.4 = 9.4 \)

Check: \( 6.02 \times 9.4 = 56.588 \). Correct. Answer: \( 9.4 \)

Part i: \( 10 \enclose{longdiv}{84.6} \)

Dividing by \( 10 \) moves decimal left one place: \( 84.6 \div 10 = 8.46 \)
Check: \( 10 \times 8.46 = 84.6 \). Correct. Answer: \( 8.46 \)

Part j: \( 7.2 \enclose{longdiv}{22.104} \)

Multiply divisor and dividend by \( 10 \): \( 72 \enclose{longdiv}{221.04} \)

• \( 72 \times 3 = 216 \), \( 221.04 - 216 = 5.04 \)
• \( 72 \times 0.07 = 5.04 \)
• Quotient: \( 3 + 0.07 = 3.07 \)

Check: \( 7.2 \times 3.07 = 22.104 \). Correct. Answer: \( 3.07 \)

Part k: \( 5.67 \enclose{longdiv}{41.391} \)

Multiply divisor and dividend by \( 100 \): \( 567 \enclose{longdiv}{4139.1} \)

• \( 567 \times 7 = 3969 \), \( 4139.1 - 3969 = 170.1 \)
• \( 567 \times 0.3 = 170.1 \)
• Quotient: \( 7 + 0.3 = 7.3 \)

Check: \( 5.67 \times 7.3 = 41.391 \). Correct. Answer: \( 7.3 \)

Final Answers:

a. \( \boldsymbol{2.022} \)
b. \( \boldsymbol{4.78} \)
c. \( \boldsymbol{9.3} \)
d. \( \boldsymbol{3.48} \)
e. \( \boldsymbol{47.34} \)
f. \( \boldsymbol{9.78} \)
g. \( \boldsymbol{2.63} \)
h. \( \boldsymbol{9.4} \)
i. \( \boldsymbol{8.46} \)
j. \( \boldsymbol{3.07} \)
k. \( \boldsymbol{7.3} \)