Question

solve the problems.
1 simplify 18.1 − 1.1(7.4 − 9.6 + 0.5). write your answers in the blanks.
18.1 − 1.1(7.4 − 9.6 + 0.5) = 18.1 − 1.1(_______ + 0.5)
= 18.1 − 1.1(_______)
= 18.1 + _______
= _______
2 kendra knits hats. she starts with 3 3/4 yd of yarn. after knitting for 5 1/4 h, there is
1 1/8 yd of yarn left. at what rate does the amount of yarn change? show your work.
solution
3 kevin has a checking account. this month he deposits three paychecks. each
paycheck is for $225.75. he also makes two payments for utilities. each payment
is for $75. he makes a third payment for $125. what is the approximate change
in kevin’s account? show your work.

Explanation:

Problem 1

Step 1: Distribute the -1.1

Using the distributive property \(a(b + c + d)=ab+ac + ad\), here \(a=- 1.1\), \(b = 7.4\), \(c=-9.6\), \(d = 0.5\). So \(18.1-1.1(7.4 - 9.6+0.5)=18.1-1.1\times7.4+1.1\times9.6 - 1.1\times0.5=18.1-8.14 + 10.56-0.55\) (first blank: \(8.14\), second blank: \(- 9.6\), third blank: \(0.55\))

Step 2: Simplify the parentheses

First, calculate inside the parentheses: \(7.4-9.6 + 0.5=7.4 + 0.5-9.6=7.9 - 9.6=- 1.7\)

Step 3: Multiply -1.1 with -1.7

\(18.1-1.1\times(-1.7)=18.1 + 1.87\) (fourth blank: \(1.87\))

Step 4: Add 18.1 and 1.87

\(18.1+1.87 = 19.97\) (fifth blank: \(19.97\))

Problem 2

Step 1: Find the amount of yarn used

The initial amount of yarn is \(3\frac{3}{4}=\frac{15}{4}\) yd, the remaining amount is \(1\frac{1}{8}=\frac{9}{8}\) yd. The amount of yarn used \(y=\frac{15}{4}-\frac{9}{8}=\frac{30 - 9}{8}=\frac{21}{8}\) yd.

Step 2: Calculate the rate

The time taken is \(5\frac{1}{4}=\frac{21}{4}\) h. The rate \(r=\frac{\text{Amount of yarn used}}{\text{Time}}=\frac{\frac{21}{8}}{\frac{21}{4}}=\frac{21}{8}\times\frac{4}{21}=\frac{1}{2}\) yd per hour.

Problem 3

Step 1: Calculate total deposits

Kevin deposits three paychecks each of \(\$225.75\). Total deposit \(D = 3\times225.75=\$677.25\)

Step 2: Calculate total payments

He makes two payments of \(\$75\) and one payment of \(\$125\). Total payment \(P=2\times75 + 125=150 + 125=\$275\)

Step 3: Calculate the change in account balance

The change in balance \(\Delta=D - P=677.25-275=\$402.25\) (approximate value can also be calculated by rounding \(225.75\approx226\), \(D\approx3\times226 = 678\), \(P\approx2\times75+125 = 275\), \(\Delta\approx678 - 275=\$403\))

Final Answers

Problem 1 Answers:

First blank: \(8.14\), Second blank: \(-9.6\), Third blank: \(0.55\), Fourth blank: \(1.87\), Fifth blank: \(19.97\)

Problem 2 Answer:

The rate at which the amount of yarn changes is \(\frac{1}{2}\) yd per hour (or \(0.5\) yd per hour)

Problem 3 Answer:

The approximate change in Kevin's account is \(\$402.25\) (or approximately \(\$403\))

Answer:

Problem 1

Step 1: Distribute the -1.1

Using the distributive property \(a(b + c + d)=ab+ac + ad\), here \(a=- 1.1\), \(b = 7.4\), \(c=-9.6\), \(d = 0.5\). So \(18.1-1.1(7.4 - 9.6+0.5)=18.1-1.1\times7.4+1.1\times9.6 - 1.1\times0.5=18.1-8.14 + 10.56-0.55\) (first blank: \(8.14\), second blank: \(- 9.6\), third blank: \(0.55\))

Step 2: Simplify the parentheses

First, calculate inside the parentheses: \(7.4-9.6 + 0.5=7.4 + 0.5-9.6=7.9 - 9.6=- 1.7\)

Step 3: Multiply -1.1 with -1.7

\(18.1-1.1\times(-1.7)=18.1 + 1.87\) (fourth blank: \(1.87\))

Step 4: Add 18.1 and 1.87

\(18.1+1.87 = 19.97\) (fifth blank: \(19.97\))

Problem 2

Step 1: Find the amount of yarn used

The initial amount of yarn is \(3\frac{3}{4}=\frac{15}{4}\) yd, the remaining amount is \(1\frac{1}{8}=\frac{9}{8}\) yd. The amount of yarn used \(y=\frac{15}{4}-\frac{9}{8}=\frac{30 - 9}{8}=\frac{21}{8}\) yd.

Step 2: Calculate the rate

The time taken is \(5\frac{1}{4}=\frac{21}{4}\) h. The rate \(r=\frac{\text{Amount of yarn used}}{\text{Time}}=\frac{\frac{21}{8}}{\frac{21}{4}}=\frac{21}{8}\times\frac{4}{21}=\frac{1}{2}\) yd per hour.

Problem 3

Step 1: Calculate total deposits

Kevin deposits three paychecks each of \(\$225.75\). Total deposit \(D = 3\times225.75=\$677.25\)

Step 2: Calculate total payments

He makes two payments of \(\$75\) and one payment of \(\$125\). Total payment \(P=2\times75 + 125=150 + 125=\$275\)

Step 3: Calculate the change in account balance

The change in balance \(\Delta=D - P=677.25-275=\$402.25\) (approximate value can also be calculated by rounding \(225.75\approx226\), \(D\approx3\times226 = 678\), \(P\approx2\times75+125 = 275\), \(\Delta\approx678 - 275=\$403\))

Final Answers

Problem 1 Answers:

First blank: \(8.14\), Second blank: \(-9.6\), Third blank: \(0.55\), Fourth blank: \(1.87\), Fifth blank: \(19.97\)

Problem 2 Answer:

The rate at which the amount of yarn changes is \(\frac{1}{2}\) yd per hour (or \(0.5\) yd per hour)

Problem 3 Answer:

The approximate change in Kevin's account is \(\$402.25\) (or approximately \(\$403\))