Question

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.

Explanation:

Translate and model each word problem

Using the Translating Word Problems and Algebraic Modeling knowledge points

• Question 1: Let the number be \(x\). "Five less than twice some number is equal to 21" translates to:

$$2x - 5 = 21$$

• Question 2: Let the width be \(w\) feet. The length is \(2w\). The perimeter is \(2(\text{length} + \text{width}) = 36\):

$$2(2w + w) = 36 \implies 6w = 36$$

• Question 3: Let Tony's age be \(t\). Lenny is \(t + 4\). Rita is \(2(t + 4)\). The sum of their ages is 72:

$$t + (t + 4) + 2(t + 4) = 72 \implies 4t + 12 = 72$$

• Question 4: Let the number of nickels be \(n\). The number of dimes is \(n + 2\). The total value in cents is 185:

$$5n + 10(n + 2) = 185 \implies 15n + 20 = 185$$

• Question 5: Let the first even integer be \(x\). The next consecutive even integer is \(x + 2\). Their sum is \(-22\):

$$x + (x + 2) = -22 \implies 2x + 2 = -22$$

• Question 6: Let the number be \(x\). "Twice some number decreased by 5 is equal to 1 more than that number" translates to:

$$2x - 5 = x + 1$$

• Question 7: Let the three consecutive integers be \(n\), \(n+1\), and \(n+2\). Their sum is 102:

$$n + (n + 1) + (n + 2) = 102 \implies 3n + 3 = 102$$

• Question 8: Let the number of \(\$5\) bills be \(f\). The number of \(\$1\) bills is \(2f\). The total value of these bills is \(\$31 - \$10 = \$21\):

$$5f + 1(2f) = 21 \implies 7f = 21$$

• Question 9: Let the first odd integer be \(x\). The next consecutive odd integer is \(x + 2\). Their sum is 0:

$$x + (x + 2) = 0 \implies 2x + 2 = 0$$

• Question 10: Let the cost of the multimeter be \(m\) dollars. The power monitor costs \(m + 44.04\). Together they cost \(417.94\):

$$m + (m + 44.04) = 417.94 \implies 2m + 44.04 = 417.94$$

Solve the linear equations

Using the Linear Equations Solving knowledge point

• Question 1:

$$2x = 26 \implies x = 13$$

• Question 2:

$$6w = 36 \implies w = 6$$

• Question 3:

$$4t = 60 \implies t = 15$$

Rita's age is \(2(t + 4) = 2(19) = 38\).

• Question 4:

$$15n = 165 \implies n = 11$$

Nickels: 11, Dimes: \(11 + 2 = 13\).

• Question 5:

$$2x = -24 \implies x = -12$$

The integers are \(-12\) and \(-10\).

• Question 6:

$$2x - x = 1 + 5 \implies x = 6$$

• Question 7:

$$3n = 99 \implies n = 33$$

The integers are 33, 34, and 35.

• Question 8:

$$7f = 21 \implies f = 3$$

The number of \(\$1\) bills is \(2f = 2(3) = 6\).

• Question 9:

$$2x = -2 \implies x = -1$$

The integers are \(-1\) and \(1\).

• Question 10:

$$2m = 373.90 \implies m = 186.95$$

Multimeter: \(\$186.95\), Power monitor: \(186.95 + 44.04 = \$230.99\).

Answer:

Question 1

The number is 13.

Question 2

The width is 6 feet.

Question 3

Rita is 38 years old.

Question 4

There are 11 nickels and 13 dimes.

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 multimeter costs \(\$186.95\) and the power monitor costs \(\$230.99\).