Written solutions to the chapter problems. Check each step, then correct yourself.
1 Sign rules ★★★
a) different signs: \(-30\). b) same signs: \(24\). c) different signs: \(-63\). d) three negative factors, odd, so \(-1\).
2 Dividing integers ★★★
a) \(-8\). b) \(8\). c) \(-5\). d) \(0\), since zero divided by a nonzero number is zero.
3 Missing numbers ★★★
a) \(35\div(-5)=-7\). b) \(9\cdot(-4)=-36\). c) \(72\div(-8)=-9\). Each answer can be checked by multiplying back.
4 Multiplying fractions ★★★
a) \(\dfrac{6}{35}\). b) \(\dfrac{24}{60}=\dfrac{2}{5}\). c) different signs: \(-\dfrac{6}{14}=-\dfrac{3}{7}\).
5 Reciprocals ★★★
a) \(\dfrac{8}{3}\). b) \(-\dfrac{1}{5}\). c) \(2\dfrac{1}{2}=\dfrac{5}{2}\), so the reciprocal is \(\dfrac{2}{5}\).
6 Fractions to decimals ★★★
a) \(3\div 8=0.375\). b) \(7\div 20=0.35\). c) \(5\div 6=0.8\overline{3}\). d) \(2\div 9=0.\overline{2}\).
7 True or false? ★★★
a) False: same signs give a positive product, for example \((-2)(-3)=6\). b) True: same signs give a positive quotient, \(4\). c) False: the reciprocal keeps the sign, so it is \(-\dfrac{3}{2}\).
8 Milk for pancakes ★★★
\(\dfrac{3}{4}\cdot 2\dfrac{2}{3}=\dfrac{3}{4}\cdot\dfrac{8}{3}=\dfrac{24}{12}=2\). The batches need 2 liters of milk.
9 Mixed number division ★★★
a) \(\dfrac{7}{2}\cdot\dfrac{4}{7}=\dfrac{28}{14}=2\). b) \(-\dfrac{12}{5}\cdot\dfrac{10}{3}=-\dfrac{120}{15}=-8\).
10 Integer order of operations ★★★
a) Parentheses: \(-5+2=-3\). Exponent: \((-3)^2=9\). Then \(4\cdot 9=36\) and \(-3+36=33\). b) Left to right: \(18\div(-3)=-6\), \(-6\cdot 2=-12\), then \(-12-(-4)=-8\).
11 Terminate or repeat? ★★★
Each fraction is in lowest terms. \(8=2^3\) and \(25=5^2\), so \(\dfrac{7}{8}\) and \(\dfrac{11}{25}\) terminate. \(12=2^2\cdot 3\) and \(21=3\cdot 7\) contain other primes, so \(\dfrac{5}{12}\) and \(\dfrac{4}{21}\) repeat.
12 Repeating blocks ★★★
a) \(0.2\overline{6}\). b) \(0.\overline{714285}\), a block of 6 digits. c) \(0.58\overline{3}\). d) \(0.\overline{36}\).
13 Distributive shortcuts ★★★
a) \(15\cdot\left(-\dfrac{2}{5}\right)+15\cdot\dfrac{1}{3}=-6+5=-1\). b) \(24\cdot\dfrac{5}{6}-24\cdot\dfrac{3}{8}=20-9=11\).
14 Freezer ★★★
Total change: \(6\cdot(-2.5)=-15\). Final: \(14+(-15)=-1\). The freezer ends at \(-1^\circ\)F.
15 Find the error ★★★
The student multiplied instead of using the reciprocal of the divisor. Correct: \(-\dfrac{2}{3}\cdot\dfrac{9}{4}=-\dfrac{18}{12}=-\dfrac{3}{2}\).
16 Name the property ★★★
a) commutative. b) associative. c) identity. d) multiplicative inverse (reciprocals). e) distributive.
17 Bank balance ★★★
Change: \(7\cdot(-18.50)=-129.50\). Balance: \(60-129.50=-69.50\). The account is overdrawn by 69.50 dollars.
18 Rain barrel ★★★
Water in the barrel: \(60\cdot\dfrac{5}{6}=50\) gallons. Poured out: \(50\cdot\dfrac{3}{10}=15\) gallons. Buckets: \(15\div 2\dfrac{1}{2}=15\cdot\dfrac{2}{5}=6\). Six buckets are filled.
19 Mean of changes ★★★
Sum: \(-3.5+2-1.25+0.75-4=-6\). Mean: \(-6\div 5=-1.2\). The mean is \(-1.2^\circ\)F.
20 A big expression ★★★
a) Top: \(-\dfrac{9}{12}+\dfrac{10}{12}=\dfrac{1}{12}\). Then \(\dfrac{1}{12}\div\left(-\dfrac{1}{3}\right)=\dfrac{1}{12}\cdot(-3)=-\dfrac{1}{4}\). b) \(\dfrac{5}{2}\cdot\left(-\dfrac{4}{5}\right)=-2\); \((-3)^2=9\) and \(9\div(-6)=-1.5\). So \(-2-(-1.5)=-0.5\).
21 Odd and even ★★★
a) 15 is odd so \((-1)^{15}=-1\); 20 is even so \((-1)^{20}=1\); \((-2)^5=-32\). b) There are 7 negative factors, an odd count, so the product is negative. The absolute values multiply to \(2\cdot 3\cdot 1\cdot 5\cdot 1\cdot 2\cdot 4=240\). The product is \(-240\).
22 Ordering mixed forms ★★★
\(0.7\); \(\dfrac{5}{7}=0.714285\ldots\); \(\dfrac{8}{11}=0.7272\ldots\); \(0.\overline{7}=0.7777\ldots\). Order: \(0.7<\dfrac{5}{7}<\dfrac{8}{11}<0.\overline{7}\).
23 Reading rate ★★★
a) \(3\dfrac{1}{2}\div\dfrac{1}{4}=\dfrac{7}{2}\cdot 4=14\) pages per hour. b) \(49\div 14=\dfrac{7}{2}=3\dfrac{1}{2}\). The chapter takes 3.5 hours, or 3 hours 30 minutes.
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