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.

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{ "quality": "clear", "question_count": 10, "skills_matched": [ "step_cot", "explain_then_answer", "tutor" ], "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 **Algebraic Modeling** knowledge points: Let the number be $x$. "Five less than twice some number" translates to $2x - 5$. "is equal to 21" translates to $2x - 5 = 21$. Using the **Linear Equations Solving** knowledge point: $$ \begin{aligned} 2x - 5 &= 21 \ 2x &= 26 \ x &= 13 \end{aligned} $$ ## Translate and solve Question 2 Using the **Translating Word Problems** and **Algebraic Modeling** knowledge points: Let the width be $w$ feet. The length is $2w$ feet. The perimeter of a rectangle is given by $2(\text{length} + \text{width}) = 36$. $$2(2w + w) = 36$$ Using the **Linear Equations Solving** knowledge point: $$ \begin{aligned} 2(3w) &= 36 \ 6w &= 36 \ w &= 6 \end{aligned} $$ ## Translate and solve Question 3 Using the **Translating Word Problems** and **Algebraic Modeling** knowledge points: Let Tony's age be $t$. Lenny is 4 years older than Tony: $l = t + 4$. Rita is twice as old as Lenny: $r = 2(t + 4) = 2t + 8$. The sum of all three ages is 72: $$t + (t + 4) + (2t + 8) = 72$$ Using the **Linear Equations Solving** knowledge point: $$ \begin{aligned} 4t + 12 &= 72 \ 4t &= 60 \ t &= 15 \end{aligned} $$ Rita's age is $r = 2(15) + 8 = 38$. ## Translate and solve Question 4 Using the **Translating Word Problems** and **Algebraic Modeling** knowledge points: Let the number of nickels be $n$. The number of dimes is $n + 2$. The total value in cents is $5n + 10(n + 2) = 185$. Using the **Linear Equations Solving** knowledge point: $$ \begin{aligned} 5n + 10n + 20 &= 185 \ 15n &= 165 \ n &= 11 \end{aligned} $$ Nickels: 11, Dimes: $11 + 2 = 13$. ## Translate and solve Question 5 Using the **Translating Word Problems** and **Algebraic Modeling** knowledge points: Let the first even integer be $x$. The next consecutive even integer is $x + 2$. Their sum is $-22$: $$x + (x + 2) = -22$$ Using the **Linear Equations Solving** knowledge point: $$ \begin{aligned} 2x + 2 &= -22 \ 2x &= -24 \ x &= -12 \end{aligned} $$ The two integers are $-12$ and $-10$. ## Translate and solve Question 6 Using the **Translating Word Problems** and **Algebraic Modeling** knowledge points: Let the number be $x$. "Twice some number decreased by 5" is $2x - 5$. "1 more than that number" is $x + 1$. $$2x - 5 = x + 1$$ Using the **Linear Equations Solving** knowledge point: $$ \begin{aligned} 2x - x &= 1 + 5 \ x &= 6 \end{aligned} $$ ## Translate and solve Question 7 Using the **Translating Word Problems** and **Algebraic Modeling** knowledge points: Let the three consecutive integers be $x$, $x + 1$, and $x + 2$. Their sum is 102: $$x + (x + 1) + (x + 2) = 102$$ Using the **Linear Equations Solving** knowledge point: $$ \begin{aligned} 3x + 3 &= 102 \ 3x &= 99 \ x &= 33 \end{aligned} $$ The integers are 33, 34, and 35. ## Translate and solve Question 8 Using the **Translating Word Problems** and **Algebraic Modeling** knowledge points: Let the number of \$5 bills be $f$. The number of \$1 bills is $2f$. Xavier has one \$10 bill. The total value is $10(1) + 1(2f) + 5(f) = 31$. Using the **Linear Equations Solving** knowledge point: $$ \begin{aligned} 10 + 2f + 5f &= 31 \ 7f &= 21 \ f &= 3 \end{aligned} $$ The number of \$1 bills is $2(3) = 6$. ## Translate and solve Question 9 Using the **Translating Word Problems** and **Algebraic Modeling** knowledge points: Let the first odd integer be $x$. The next consecutive odd integer is $x + 2$. Their sum is 0: $$x + (x + 2) = 0$$ Using the **Linear Equations Solving** knowledge point: $$ \begin{aligned} 2x + 2 &= 0 \ 2x &= -2 \ x &= -1 \end{aligned} $$ The two integers are $-1$ and $1$. ## Translate and solve Question 10 Using the **Translating Word Problems** and **Algebraic Modeling** knowledge points: Let the cost of the multimeter be $m$ dollars. The power monitor costs $m + 44.04$ dollars. Together they cost 417.94 dollars: $$m + (m + 44.04) = 417.94$$ Using the **Linear Equations Solving** knowledge point: $$ \begin{aligned} 2m + 44.04 &= 417.94 \ 2m &= 373.90 \ m &= 186.95 \end{aligned} $$ Multimeter: \$186.95, Power monitor: $186.95 + 44.04 = 230.99$ dollars. ## Question 1 The number is 13. ## Question 2 The width is 6 feet. ## Question 3 Rita is 38 years old. ## Question 4 There are 13 dimes and 11 nickels. ## Question 5 The two integers are $-12$ and $-10$. ## Question 6 The number is 6. ## Question 7 The three consecutive integers are 33, 34, and 35. ## Question 8 There are 6 \$1 bills. ## Question 9 The two consecutive odd integers are $-1$ and 1. ## Question 10 The digital multimeter costs \$186.95 and the portable power monitor costs \$230.99. { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Algebra", "Translating Word Problems" ] }

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