exercises 10-5
solve each word problem.
1. five less than twice some number is equal to 21. what is the number?
2. if the perimeter of a certain rectangle is 36 feet and the length is 2 times the width, what is the width?
3. rita is twice as old as her cousin lenny, who is 4 years older than their cousin tony. the sum of all three ages is 72. how old is rita?
4. a handful of coins is made up of dimes and nickels. there are 2 more dimes than nickels. the coins have a value of $1.85. how many of each coin are there?
5. the sum of two consecutive even integers is -22. what are the two integers?
6. twice some number decreased by 5 is equal to 1 more than that number. what is the number?
7. the sum of three consecutive integers is 102. what are the three consecutive integers?
8. xavier has one $10 bill, some $1 bills, and some $5 bills for a total of $31. there are twice as many $1 bills as $5 bills. how many $1 bills are there?
9. the sum of two consecutive odd integers is zero. what are the two odd integers?
10. hilkka purchases a digital multimeter and a portable power monitor. together they cost $417.94. the power monitor costs $44.04 more than the multimeter. find the cost of each tool.

Question

Accepted Answer

{ "quality": "clear", "question_count": 10, "skills_matched": [ "step_cot", "table_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Translating Word Problems", "Algebraic Modeling", "Linear Equations Solving" ], "new_concepts": [], "current_concepts": [ "Translating Word Problems", "Algebraic Modeling", "Linear Equations Solving" ] } ## Translate and solve Question 1 Using the *Translating Word Problems* and *Linear Equations Solving* knowledge points $$ \begin{aligned} &2x - 5 = 21\ &2x = 26\ &x = 13 \end{aligned} $$ ## Translate and solve Question 2 Using the *Algebraic Modeling* and *Linear Equations Solving* knowledge points $$ \begin{aligned} &P = 2(l + w) = 36\ &l = 2w\ &2(2w + w) = 36 \implies 6w = 36 \implies w = 6 \end{aligned} $$ ## Translate and solve Question 3 Using the *Translating Word Problems* and *Linear Equations Solving* knowledge points $$ \begin{aligned} &R = 2L,\quad L = T + 4 \implies T = L - 4\ &R + L + T = 72 \implies 2L + L + (L - 4) = 72\ &4L - 4 = 72 \implies 4L = 76 \implies L = 19\ &R = 2(19) = 38 \end{aligned} $$ ## Translate and solve Question 4 Using the *Algebraic Modeling* and *Linear Equations Solving* knowledge points $$ \begin{aligned} &d = n + 2\ &0.10d + 0.05n = 1.85 \implies 10d + 5n = 185\ &10(n + 2) + 5n = 185 \implies 15n + 20 = 185\ &15n = 165 \implies n = 11\ &d = 11 + 2 = 13 \end{aligned} $$ ## Translate and solve Questions 5 to 10 Using the *Translating Word Problems* and *Linear Equations Solving* knowledge points $$ \begin{aligned} &\text{Q5: } x + (x + 2) = -22 \implies 2x = -24 \implies x = -12, \text{ integers are } -12, -10\ &\text{Q6: } 2x - 5 = x + 1 \implies x = 6\ &\text{Q7: } x + (x + 1) + (x + 2) = 102 \implies 3x + 3 = 102 \implies 3x = 99 \implies x = 33, \text{ integers are } 33, 34, 35\ &\text{Q8: } 10 + 1y + 5z = 31 \implies y + 5z = 21\ &y = 2z \implies 2z + 5z = 21 \implies 7z = 21 \implies z = 3 \implies y = 6\ &\text{Q9: } x + (x + 2) = 0 \implies 2x = -2 \implies x = -1, \text{ integers are } -1, 1\ &\text{Q10: } M + P = 417.94,\quad P = M + 44.04\ &M + (M + 44.04) = 417.94 \implies 2M = 373.90 \implies M = 186.95\ &P = 186.95 + 44.04 = 230.99 \end{aligned} $$ | No. | Answer | |---|---| | 1 | $13$ | | 2 | $6\text{ feet}$ | | 3 | $38\text{ years old}$ | | 4 | $11\text{ nickels, } 13\text{ dimes}$ | | 5 | $-12\text{ and } -10$ | | 6 | $6$ | | 7 | $33, 34, 35$ | | 8 | $6$ | | 9 | $-1, 1$ | | 10 | $\text{Multimeter: }\$186.95\text{, Power monitor: }\$230.99$ | { "subject": "Mathematics", "question_type": "Word Problem", "knowledge_point": [ "Mathematics", "Algebra", "Translating Word Problems" ] }

No.	Answer
2	\(6\text{ feet}\)
3	\(38\text{ years old}\)
4	\(11\text{ nickels, } 13\text{ dimes}\)
5	\(-12\text{ and } -10\)
6	\(6\)
7	\(33, 34, 35\)
8	\(6\)
9	\(-1, 1\)
10	\(\text{Multimeter: }\$186.95\text{, Power monitor: }\$230.99\)

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