Supplementary Exercises

Section 3.5 Supplementary Exercises

Subsection 3.5.1 Number Systems Questions

Exercises Exercises

1.

What is the value (in base ten) of the underlined digit in the following numbers?

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$\displaystyle 11\underline{1}00011_{two}$

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$\displaystyle 6\underline{3}72_{eleven}$

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$\displaystyle 320\underline{2}10_{four}$

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$\displaystyle 40\underline{8}063_{twelve}$

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$\displaystyle \underline{5}025_{seven}$

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$\displaystyle 8100\underline{5}_{ten}$

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2.

Roman numerals. The symbols I, V, X, L, C, D, M represent 1, 5, 10, 50, 100, 500, 1000. Numerals are written from highest to lowest (left to right) and summed, except when a smaller numeral precedes a larger one, indicating subtraction (e.g., IV = 5 − 1 = 4). Convert to base ten:

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XV

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CMLXXXVII

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XXIX

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XLIV

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DCCLXVIII

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MDCCCXII

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3.

Convert each of the following to base ten.

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$\displaystyle 12_{eleven}$

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$\displaystyle 3122_{four}$

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$\displaystyle 615_{seven}$

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$\displaystyle 1005_{six}$

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$\displaystyle 2703_{nine}$

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$\displaystyle 110100_{two}$

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$\displaystyle 13A48_{twelve}$

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$\displaystyle 945_{twenty}$

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4.

Consider a time-like mixed-base system (seconds–minutes–hours–days). Use examples to demonstrate:

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$0-0-0-60 = 0-0-1-0_{time}\text{,}$ $0-0-0-3600 = 1-0-0_{time}\text{.}$

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$1234$ meaning “1 day, 2 hours, 3 minutes, 4 seconds.” Convert this to a single base-ten count (seconds).

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5.

Order each list from least to greatest.

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$\displaystyle 3_{nine},\; 11_{eight},\; 300_{six},\; 42_{five},\; 92_{eleven}$

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$\displaystyle 301_{five},\; 302_{four},\; 81_{eleven},\; 10002_{three},\; 450_{twelve}$

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$\displaystyle 61_{eleven},\; 122_{seven},\; 64_{ten},\; 111111_{two},\; 1002_{four}$

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$\displaystyle 1034_{five},\; 331_{seven},\; 121_{nine},\; 69_{twelve},\; 321_{six}$

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$\displaystyle 101111_{two},\; 38_{eleven},\; 45_{eight},\; 111_{six},\; 133_{four}$

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$\displaystyle 56_{eleven},\; 114_{five},\; 110100_{two},\; 43_{ten},\; 31_{eight}$

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6.

Each list is ordered from least to greatest. Find a possible value for each missing base so the ordering is correct (bases are integers large enough to permit the shown digits).

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$\displaystyle 10_{three} \lt 12_{a} \lt 20_{five} \lt 24_{six}$

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$\displaystyle 101_{two} \lt 22_{b} \lt 24_{eight} \lt 30_{seven}$

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$\displaystyle 100_{four} \lt 1A_{eleven} \lt 33_{c} \lt 40_{nine}$

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$\displaystyle 1001_{two} \lt 210_{d} \lt 122_{five} \lt 51_{eight}$

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$\displaystyle 7_{ten} \lt 20_{e} \lt 111_{three} \lt 1A_{twelve}$

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$\displaystyle 222_{four} \lt 10000_{three} \lt 210_{seven} \lt 121_{f}$

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7.

Fill in the missing symbol ($<\text{,}$ $>\text{,}$ or $=$) between each pair.

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$\displaystyle 56_{eight}\; \square \; 63_{seven}$

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$\displaystyle 1B_{twelve}\; \square \; 120_{four}$

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$\displaystyle 71_{nine}\; \square \; 144_{six}$

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$\displaystyle 90_{eleven}\; \square \; 1000110_{two}$

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$\displaystyle 2122_{three}\; \square \; 203_{five}$

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$\displaystyle 101010_{two}\; \square \; 2A_{sixteen}$

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8.

Identify the missing base in each equation.

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$\displaystyle 9_{twelve} + 10_{four} = 21_{\square}$

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$\displaystyle 31_{five} + 17_{nine} = 200_{\square}$

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$\displaystyle 1020_{three} - 14_{eight} = 21_{\square}$

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$\displaystyle 82_{\square} + 35_{seven} = 400_{five}$

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$\displaystyle 57_{eight} - 30_{five} = 112_{\square}$

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$\displaystyle 46_{twelve} + 1203_{\square} = 12A_{eleven}$

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9.

Is a “base one” number system possible? If so, describe it. If not, which place-value rules does it violate, and how might you adjust them to make a unary system usable?

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10.

Consider the following (non-standard) place-value system (places increase from right to left):

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1st place: ones;

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2nd place: up to four 1st places (so one 2nd place = five 1st places);

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3rd place: up to seven 2nd places (so one 3rd place = eight 2nd places);

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4th place: up to ten 3rd places; and so on.

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Digits beyond 9 use uppercase letters: $A=10\text{,}$ $B=11\text{,}$ $C=12\text{,}$ etc.

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Convert to base ten:
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1. 23
  
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2. 5633
  
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3. AB123
  
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4. 1B20A
  
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Convert the base-ten numbers to this system:
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1. 97
  
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2. 2021
  
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3. 5000
  
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11.

Do tally marks fit our definition of a place-value number system? Justify your answer.

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12.

Convert each number to the indicated base.

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$345_{ten}$ to base $7$

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$1001101_{two}$ to base $8$

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$7A5_{sixteen}$ to base $9$

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$1234_{five}$ to base $10$

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$888_{nine}$ to base $3$

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$2024_{ten}$ to base $12$

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13.

Create your own number system: specify the digit set, the place-value rule (fixed base or mixed-base), and demonstrate how to write and interpret at least five example numbers. Include one example of addition in your system.

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14.

Find $x$ so that the equality is true (specify $x$ in the named base).

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$\displaystyle 401_{ten} = x_{three}$

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$\displaystyle 2A_{sixteen} = x_{ten}$

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$\displaystyle x_{five} = 324_{seven}$

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$\displaystyle 100000_{two} = x_{ten}$

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$\displaystyle x_{twelve} = 999_{ten}$

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$\displaystyle 731_{eight} = x_{four}$

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15.

Count on your fingers in base two. What is the largest number you can represent? What about base three? Base four?

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16.

Fill the blank with a number written in the indicated base so that the statement is true.

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$\displaystyle 100_{two} \lt \square_{three} \lt 22_{five}$

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$\displaystyle 17_{eight} \lt \square_{four} \lt 24_{seven}$

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$\displaystyle 2A_{twelve} \lt \square_{ten} \lt 1111_{two}$

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$\displaystyle 33_{six} \lt \square_{nine} \lt 40_{five}$

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$\displaystyle 1010_{two} \lt \square_{eight} \lt 31_{five}$

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$\displaystyle 90_{eleven} \lt \square_{sixteen} \lt 400_{seven}$

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Subsection 3.5.2 Arithmetic Operations: Practice

Exercises Exercises

1.

Compute mentally.

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$\displaystyle 398 + 257$

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$\displaystyle 250 + 175 + 325$

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$\displaystyle 1{,}999 + 3{,}501$

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2.

Compute each difference (you may use a “same-change” strategy).

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$\displaystyle 1{,}000 - 398$

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$\displaystyle 10{,}003 - 4{,}999$

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$\displaystyle 5{,}004 - 2{,}997$

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3.

Circle the larger value.

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$37 \times 48$ or $36 \times 49$

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$501 - 248$ or $500 - 249$

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$25 \times 32$ or $20 \times 37$

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4.

Compute using the distributive property (partial products).

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$\displaystyle 27 \times 16$

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$\displaystyle 304 \times 25$

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$\displaystyle 99 \times 48$

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5.

Evaluate.

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\begin{gather*} 48 + (17 + 12)\\ (25 \times 4) \times 5\\ 9 \times (a + b) \;\text{ with } a=6,\; b=8\\ 64 + 0\\ 0 \times 735 \end{gather*}

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6.

Compute.

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$\displaystyle 3^4 \times 3^2$

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$\displaystyle 10^3 + 10^3$

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$\displaystyle 5^0 + 4^0$

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7.

Find the quotient $q$ and remainder $r$ (so $a = bq + r$ with $0 \le r < b$).

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$\displaystyle 587 \div 23$

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$\displaystyle 127 \div 12$

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$\displaystyle 1{,}005 \div 32$

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8.

A bus holds 42 students.

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There are 389 students. How many full buses are needed? How many students ride on the last bus?

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If every bus must be either full or empty, how many buses are required for 389 students?

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9.

Compute using the distributive property.

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$\displaystyle 25 \times (20 - 2)$

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$\displaystyle (60 + 7)\times 14$

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10.

Compute mentally.

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$\displaystyle 125 \times 32$

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$\displaystyle 50 \times 47$

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$\displaystyle 24 \times 75$

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11.

Estimate, then compute exactly.

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$\displaystyle 683 \times 29$

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$\displaystyle 1{,}497 \times 51$

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$\displaystyle 198 \times 203$

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12.

Compute.

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\begin{gather*} 37 + 25 + 63 + 75\\ 250 \times 16\\ (300 - 1)\times 48 \end{gather*}

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13.

Find the missing number.

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$\displaystyle \square + 47 = 123$

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$\displaystyle 9 \times \square = 468$

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$\displaystyle \square \div 12 = 58 \text{ remainder } 7$

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14.

For each pair $(a,b)$ with $b > 0\text{,}$ find integers $q$ and $r$ such that $a = bq + r$ and $0 \le r < b\text{.}$

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$\displaystyle (a,b) = (1{,}375, 24)$

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$\displaystyle (a,b) = (940, 45)$

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$\displaystyle (a,b) = (2{,}019, 100)$

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Subsection 3.5.3 Properties of Arithmetic: Practice

Exercises Exercises

1.

Evaluate (use standard order of operations).

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$\displaystyle 3 + 4(5 - 2)$

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$\displaystyle (18 - 6) \div 3 + 7$

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$\displaystyle 2(9 + 8) - 3^2$

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$\displaystyle 40 \div (5 \cdot 2) + 6$

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2.

State whether the set is closed under the operation. If not, give a counterexample.

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$\mathbb{N}_0$ under subtraction

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$\mathbb{Z}$ under division

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$\mathbb{Q}$ under multiplication

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$\mathbb{N}_0$ under addition

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3.

Use commutativity to compute mentally.

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$\displaystyle 27 + 58 = \square$

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$\displaystyle 58 + 27 = \square$

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$\displaystyle 14 \times 25 = \square$

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$\displaystyle 25 \times 14 = \square$

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4.

Regroup using associativity to make the computation easier.

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$\displaystyle (37 + 63) + 25$

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$\displaystyle 8 \times (25 \times 4)$

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$\displaystyle (125 + 275) + 600$

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$\displaystyle (5 \times 12) \times 25$

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5.

Name the property that justifies each equality (commutative, associative, distributive, identity, or zero property).

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\begin{gather*} 48 + (17 + 12) = (48 + 17) + 12\\ 7 \times 0 = 0\\ 9 \times (a + b) = 9a + 9b\\ 64 + 0 = 64\\ 3 \times 14 = 14 \times 3 \end{gather*}

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6.

Expand using the distributive property and simplify.

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$\displaystyle 7(30 + 6)$

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$\displaystyle (200 + 4)\cdot 19$

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$\displaystyle 9(50 - 3)$

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$(a + b)(c + d)$ (expand to four terms)

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7.

Factor using the distributive property.

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$\displaystyle 18x + 24x$

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$\displaystyle 35a + 14b$

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$\displaystyle 48 \cdot 25 + 2 \cdot 25$

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$\displaystyle 9m - 3n$

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8.

For the indicated set, give the identity element for the operation and the inverse of the given element (if it exists in the set).

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Set $\mathbb{Z}\text{,}$ operation: addition; inverse of $19$

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Set $\mathbb{Q} \setminus \{0\}\text{,}$ operation: multiplication; inverse of $\dfrac{3}{7}$

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Set $\mathbb{N}_0\text{,}$ operation: addition; inverse of $5$

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Set $\mathbb{Q}\text{,}$ operation: multiplication; inverse of $0$

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9.

Compute.

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$\displaystyle 0 \times 3{,}458$

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$\displaystyle 1 \times 3{,}458$

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$\displaystyle (a+1) - a$

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$\displaystyle 0 \cdot (a + b + c)$

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10.

Give whole-number examples showing that the property fails for the operation.

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Subtraction is not commutative.

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Division is not commutative.

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Subtraction is not associative.

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Division is not associative.

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11.

Fill in the blanks so the statements are true (whole numbers).

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$\displaystyle 87 - \square = 29$

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$\displaystyle \square - 45 = 18$

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$\displaystyle a - b = c \;\Rightarrow\; a = \square + \square$

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$\displaystyle a - c = b \;\Rightarrow\; a = \square + \square$

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12.

Write each as $a = bq + r$ with $0 \le r < b\text{.}$

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$\displaystyle 973 \div 24$

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$\displaystyle 1{,}406 \div 35$

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$\displaystyle 2{,}019 \div 100$

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13.

Compute using distribution or regrouping for efficiency.

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$\displaystyle 125 \times 32$

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$\displaystyle (300 - 1)\times 48$

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$\displaystyle 49 \times 18$

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$\displaystyle 250 \times 16$

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14.

Mark each statement True or False (whole numbers). If false, give a counterexample.

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$\displaystyle (a+b)+c = a+(b+c)$

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$\displaystyle a - b = b - a$

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$\displaystyle a(b+c) = ab + ac$

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$a \div 0$ is defined

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$0 \div a = 0$ for $a \ne 0$

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15.

Perform the indicated operation.

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Expand: $(200 + 7)(30 + 6)$

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Factor: $54x + 36y$

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Evaluate: $(18 - 6) \div 3 + 7 \cdot 2$

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Simplify: $9(a+b) - 3(a+b)$

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Subsection 3.5.4 Integers: Practice

Exercises Exercises

1.

Place on a number line and order from least to greatest.

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$\displaystyle \{-4,\,7,\,-1,\,0,\,5\}$

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$\displaystyle \{-12,\,-9,\,-15,\,-3\}$

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$\displaystyle \{8,\,-2,\,8,\,-10,\,3\}$

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2.

Compute. You may use a number-line or counter model.

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$\displaystyle 7 + (-3)$

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$\displaystyle -8 + 5$

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$\displaystyle -12 + (-9)$

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$\displaystyle 19 + (-19)$

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3.

Rewrite each difference as a sum, then compute.

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$\displaystyle 9 - 14$

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$\displaystyle -6 - 7$

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$\displaystyle -10 - (-3)$

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$\displaystyle 25 - (-12)$

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4.

Fill in the integer that makes each true.

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$\displaystyle a + \square = 0$

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$\displaystyle \square + 0 = -23$

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$\displaystyle -17 + \square = -5$

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$\displaystyle \square + (-11) = 8$

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5.

Compute.

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$\displaystyle 6\cdot(-7)$

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$\displaystyle -8\cdot-9$

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$\displaystyle -12\cdot0$

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$\displaystyle -3\cdot15$

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6.

Use the distributive property to expand or factor.

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$\displaystyle -7(12 - 3)$

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$\displaystyle (-15 + 4)\cdot 9$

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$-9x + 27x$ (factor)

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$14a - 21b$ (factor out a common integer)

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7.

Complete the steps to show $(-a) = (-1)\,a$ using distribution.

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\begin{align*} a + (-a) \amp = 0\\ -1\cdot\big(a + (-a)\big) \amp = -1\cdot 0\\ \big(-1\cdot a\big) + \big(-1\cdot(-a)\big) \amp = 0\\ \text{Therefore, } \; \square \amp = \square \end{align*}

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8.

Fill in the blanks to show $(-1)(-1)=1\text{.}$

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\begin{align*} 1 + (-1) \amp = 0\\ -1\cdot\big(1 + (-1)\big) \amp = -1\cdot 0\\ \big((-1)\cdot 1\big) + \big((-1)\cdot(-1)\big) \amp = 0\\ -1 + \square \amp = 0\\ \square \amp = 1 \end{align*}

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9.

Verify each by computing both sides (commutative/associative/distributive).

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\begin{align*} (-4)\cdot 7 \amp = 7\cdot(-4)\\ \big((-3)\cdot 5\big)\cdot 2 \amp = (-3)\cdot\big(5\cdot 2\big)\\ 6\cdot\big(8 + (-3)\big) \amp = 6\cdot 8 + 6\cdot(-3) \end{align*}

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10.

Compute the quotient (integers); if not an integer, write “not an integer.”

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$\displaystyle 84 \div -7$

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$\displaystyle -96 \div -12$

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$\displaystyle 35 \div 6$

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$\displaystyle -100 \div 25$

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11.

Answer each; show the integer arithmetic.

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At 6 AM it is $-4^\circ\text{C}$ and the temperature rises $7^\circ$ by noon. What is the noon temperature?

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A hiker is at $-120$ m and ascends $85$ m. New elevation?

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A company has a total loss of $\$60{,}000$ over 12 months. What is the average monthly loss?

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12.

Compute.

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$\displaystyle (-18) - [\,7 + (-9)\,]$

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$\displaystyle 3\cdot(-12) + 5\cdot 6$

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$\displaystyle -(2\cdot 11) + (4 - 19)$

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$\displaystyle (-3)(-8) - (27 \div -3)$

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Subsection 3.5.5 Whole Numbers: Explanatory Questions

Exercises Exercises

1.

A student says, “The digit $8$ always means eight.” Explain why this is not generally true by comparing $108_{ten}\text{,}$ $108_{nine}\text{,}$ and $108_{two}\text{.}$ In each case, state the value contributed by the 8-digit (or explain why it is not a valid digit).

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2.

Explain why $302_{five}$ and $32_{five}$ represent different numbers. Describe a blocks/regrouping picture that makes the difference unavoidable, and compute both values in base ten.

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3.

In base five, converting $77_{ten}$ gives $302_{five}\text{.}$ Explain in words why the “repeated division by 5 with remainders” algorithm produces the same digits you see when you regroup ones → longs → squares in a blocks picture.

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4.

“Same-change” subtraction: A student claims $1{,}000 - 398 = 1{,}002 - 400$ because adding $2$ to both numbers doesn’t change the answer. Do you agree? Explain why this strategy works (or when it doesn’t), using an equation or number-line argument.

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5.

Two students compute $27\times 16$ in different ways:

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\begin{gather*} \text{Student A: } 27\times(10+6)=270+162=432\\ \text{Student B: } (30-3)\times 16=480-48=432 \end{gather*}

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Explain why both methods are valid and identify the properties used in each method.

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6.

Explain why doubling–halving keeps a product the same: show that $(2a)\times \dfrac{b}{2}=ab$ whenever $b$ is even, and discuss what changes if $b$ is odd (give a specific example).

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7.

What is the difference between an identity and an inverse for an operation? Why does every integer have an additive inverse but not every integer has a multiplicative inverse (within the integers)? Give examples that include $-5\text{,}$ $0\text{,}$ and $1\text{.}$

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8.

Use an area/array (“chocolate bar”) model to justify $a(b+c)=ab+ac\text{.}$ Then extend your explanation to $(a+b)(c+d)=ac+ad+bc+bd$ as “cuts” in two directions.

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9.

Consider $0\times n$ and $n\div 0\text{.}$ Explain why the first is always $0$ (give a property or model) and the second is undefined (refer to what division would have to mean).

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10.

Two students disagree about $-12 - (-9)\text{.}$ One says “subtract a negative means the answer is negative,” the other says “subtract a negative means add the positive.” Who is right? Explain with a model (number line or chips) and compute the result.

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11.

Complete a proof that $(-1)(-1)=1$ using distribution. Fill in the blanks and explain each step in words.

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\begin{align*} 1 + (-1) \amp = 0\\ -1\cdot\big(1 + (-1)\big) \amp = -1\cdot 0\\ \big((-1)\cdot 1\big) + \big((-1)\cdot(-1)\big) \amp = 0\\ -1 + \square \amp = 0\\ \square \amp = 1 \end{align*}

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12.

A classmate orders integers by absolute value and gets: $-12 < -3 < 2 < 10$ “because $12$ is the largest absolute value.” Explain the mistake and give the correct ordering, justifying your answer.

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