Getting the Job Done—Work

Getting the Job Done—Work exercise

• Determine who has the faster work rate.

  Hints

  The unit work rate is found by dividing the numerator and denominator by the denominator.

  • The work rate is $\frac{84\text{ apples}}{4\text{ minutes}}$
  • Divide both the numerator and denominator by $4$
  • The unit work rate is $\frac{21\text{ apples}}{1\text{ minute}}$.

  Unit work rates can be compared because they both have the same denominator.

  For example: Since $21$ is greater than $18$, $\frac{21\text{ apples}}{1\text{ minute}}>\frac{18\text{ apples}}{1\text{ minute}}$

  Solution

  In the following $3$ problems, the unit work rate is found by dividing the numerator by the denominator.

  1. Holly loves pumpkins so she assigns the sisters to gather as many pumpkins from the pumpkin patch that they can.

  Matilda gathers pumpkins at a rate of $\frac{18 \text{ pumpkins}}{9 \text{ seconds}}$ and Magarete gathers at a rate of $\frac{90 \text{ pumpkins}}{30 \text{ seconds}}$.

  • Divide the numerator and denominator by $9$ to find that the unit work rate for Matilda is $\frac{2 \text{ pumpkins}}{1 \text{ second}}$.
  • Divide the numerator and denominator by $30$ to find that the unit work rate for Magarete is $\frac{3 \text{ pumpkins}}{1 \text{ second}}$.

  Since $\frac{2 \text{ pumpkins}}{1 \text{ second}}<\frac{3 \text{ pumpkins}}{1 \text{ second}}$, Matilda gathers her pumpkins at a slower rate than Magarete.

  $~$

  2. To continue on their way to see the fairy, the sisters need to rake a piles of leaves so that the path is clear.

  Matilda rakes leaves at a rate of $\frac{144 \text{ piles of leaves}}{24 \text{ minutes}}$ and Magarete rakes leaves at a rate of $\frac{155 \text{ piles of leaves}}{31 \text{ minutes}}$.

  • The unit work rate for Matilda is $\frac{6 \text{ piles of leaves}}{1 \text{ minute}}$.

  • The unit work rate for Magarete is $\frac{5 \text{ piles of leaves}}{1 \text{ minute}}$.

  Since $\frac{6 \text{ piles of leaves}}{1 \text{ minute}}>\frac{5 \text{ piles of leaves}}{1 \text{ minute}}$, Matilda rakes piles of leaves at a faster rate than Magarete.

  $~$

  3. They have almost reached the fairy but a big storm blew sticks off of a tree and the sisters need to pick them up.

  Matilda picks up sticks at a rate of $\frac{208 \text{ sticks}}{52 \text{ seconds}}$ and Magarete rakes leaves at a rate of $\frac{288 \text{ sticks}}{48 \text{ seconds}}$.

  • The unit work rate for Matilda is $\frac{4 \text{ sticks}}{1 \text{ second}}$.

  • The unit work rate for Magarete is $\frac{6 \text{ sticks}}{1 \text{ second}}$.

  Once again, since $\frac{4 \text{ sticks}}{1 \text{ second}}<\frac{6 \text{ sticks}}{1 \text{ second}}$, Matilda works at a slower rate than Magarete.

  The sisters made it to see Holly and told her about their adventure along the way.

• Order the work rates.

  Hints

  To order the values, set up the work rate as a fraction, find the unit work rate, then compare the numerators.

  A unit work rate relates a quantity to one unit of another quantity.

  Work Rate: $\frac{15 \text{ computers}}{5\text{ hours}}$

  Unit Work Rate: $\frac{3 \text{ computers}}{1\text{ hour}}$

  Solution

  1.Margarete rolls $108$ pretzels in $12$ minutes.

  • Set up the work rate as a fraction, then divide the numerator and denominator by $9$: $\frac{108\text{ pretzels}}{9\text{ minutes}}=\frac{12\text{ pretzels}}{1\text{ minute}}$

  2. Matilda rolls $50$ pretzels in $5$ minutes.

  • Set up the work rate as a fraction, then divide the numerator and denominator by $5$: $\frac{50\text{ pretzels}}{5\text{ minutes}}=\frac{10\text{ pretzels}}{1\text{ minute}}$

  3. Matilda rolls $378$ pretzels in $54$ minutes.

  • Set up the work rate as a fraction, then divide the numerator and denominator by $54$: $\frac{378\text{ pretzels}}{54\text{ minutes}}=\frac{7\text{ pretzels}}{1\text{ minute}}$

  4. Margarete rolls $252$ pretzels in $42$ minutes.

  • Set up the work rate as a fraction, then divide the numerator and denominator by $42$: $\frac{252\text{ pretzels}}{42\text{ minutes}}=\frac{6\text{ pretzels}}{1\text{ minute}}$

  5. Margarete rolls $156$ pretzels in $39$ minutes.

  • Set up the work rate as a fraction, then divide the numerator and denominator by $39$: $\frac{156\text{ pretzels}}{39\text{ minutes}}=\frac{4\text{ pretzels}}{1\text{ minute}}$

  6. Matilda rolls $57$ pretzels in $19$ minutes.

  • Set up the work rate as a fraction, then divide the numerator and denominator by $19$: $\frac{57\text{ pretzels}}{19\text{ minutes}}=\frac{3\text{ pretzels}}{1\text{ minute}}$

• Calculate the work rate for each word problem.

  Hints

  One way to find the unit rate, is to convert the rate into a fraction and divide the numerator and denominator by the denominator.

  Another way to find the unit rate, is to divide the first quantity by the second quantity so that the second quantity represents a quantity of one.

  Roy is painting a picket fence white. He can paint $26$ pickets in $2$ hours. Roy's unit rate is $13$ pickets in 1 hour.

  Solution

  1. Arthur works at his grandmother's house every weekend. Last weekend, his grandmother had him pick weeds in her yard. On Saturday, Arthur filled $28$ buckets of weeds in $4$ hours. On Sunday, Arthur filled $36$ buckets of weeds in $3$ hours.

  • To find Arthur's unit rate on Saturday, divide both numbers by $4$ to find that he can fill $7$ buckets of weeds in $1$ hour.
  • To find Arthur's unit rate on Sunday, divide both numbers by $3$ to find that he can fill $12$ buckets of weeds in $1$ hour.

  2. Margarite is opening a new boutique in town. She spends a few days building the shelves for her shoe display. Margarite built $45$ shelves in $5$ days.

  • To find Margarite's unit rate, divide both numbers by $5$ to find that she can build $9$ shelves in $1$ day.

  3. Randolph and Raul are working together on a project for school. They are required to write a $12$ page paper. Randolph writes $7$ pages in $4$ hours and Raul write $5$ pages in $2$ hours.

  • To find the unit rate at which Randolph's works, divide both numbers by $4$ to find that Randolph can write $1.75$ pages per hour.
  • The find the unit rate at which Raul works, divide both numbers by $2$ to find that Raul can write $2.5$ pages per hour.
  • To find the unit rate at which Raul and Randolph work together, add the pages together and find that they can write $4.25$ pages per hour. You do not add the hours, since the unit rates are represented by fractions and when you add fractions, only add the numerators.

• Find the fastest work rate.

  Hints

  An easier way to view and compare work rates is to write them as fractions.

  Find the unit work rate by simplifying the fraction so that there is a $1$ in the denominator.

  Since the denominator of every unit work rate is $1$, you can find the faster unit work rate by comparing numerators. The larger the numerator, the faster the unit work rate.

  Solution

  For the three problems, use the steps below to find the fastest work rate:

  1. Write all of the work rates as fraction.

  2. Find the unit work rate for each person by dividing the fraction by the denominator.

  3. The fastest work rate has the largest numerator.

  $~$

  1. Three friends, Anna, Ron, and Melia, are helping an elderly neighbor bake Christmas cookies. Anna can bake $95$ cookies in $2\frac{1}{2}$ hours, Ron can bake $72$ cookies in $2$ hours, and Melia can bake $84$ cookies in $2\frac{1}{4}$ hours.

  • Anna's work rate is, $\frac{95\text{ cookies}}{{2\frac{1}{2}}\text{ hours}}$. Divide by $2\frac{1}{2}$ to find Anna's unit work rate, $\frac{38\text{ cookies}}{1\text{ hour}}$
  • Ron's work rate is, $\frac{72\text{ cookies}}{2\text{ hours}}$. Divide by $2$ to find Ron's unit work rate, $\frac{36\text{ cookies}}{1\text{ hour}}$
  • Melia's work rate is, $\frac{84\text{ cookies}}{{2\frac{1}{4}}\text{ hours}}$. Divide by $2\frac{1}{4}$ to find Melia's unit work rate, $\frac{37.3\text{ cookies}}{1\text{ hour}}$.
  • Since $\frac{38\text{ cookies}}{1\text{ hour}}>\frac{37.3\text{ cookies}}{1\text{ hour}}>\frac{36\text{ cookies}}{1\text{ hour}}$, Anna baked cookies at the fastest rate.

  2. JC, Kate, Jade, and Mindy are competing in an apple bobbing competition. JC bobbed $20$ apples in $47$ seconds, Kate bobbed $18$ apples in $32$ seconds, Jade bobbed $24$ apples in $52$ seconds, and Mindy bobbed $15$ apples in $19$ seconds.

  • JC's work rate is, $\frac{20\text{ apples}}{47\text{ seconds}}$. Divide by $47$ to find JC's unit work rate, $\frac{0.43\text{ apples}}{1\text{ second}}$.
  • Kate's work rate is, $\frac{18\text{ apples}}{32\text{ seconds}}$. Divide by $32$ to find Kate's unit work rate, $\frac{0.56\text{ apples}}{1\text{ second}}$.
  • Jade's work rate is, $\frac{24\text{ apples}}{52\text{ seconds}}$. Divide by $52$ to find Jade's unit work rate, $\frac{0.46\text{ apples}}{1\text{ second}}$.
  • Mindy's work rate is, $\frac{15\text{ apples}}{19\text{ seconds}}$. Divide by $19$ to find Kate's unit work rate, $\frac{0.79\text{ apples}}{1\text{ second}}$.
  • Since $\frac{0.79\text{ apples}}{1\text{ second}}>\frac{0.56\text{ apples}}{1\text{ second}}>\frac{0.46\text{ apples}}{1\text{ second}}>\frac{0.43\text{ apples}}{1\text{ second}}$, Mindy bobs for apples at the fastest rate.

  3. Mrs. Fritz is needs to hire a lawn mowing company to mow her lawn. She is looking for the fastest service. D&B Lawn Mowing can mow $5$ lawns in $6\frac{1}{2}$ hours, Richard's Landscape Service can mow $7$ lawns in $8\frac{3}{4}$ hours, and Sunrise Landscape can mow $6$ lawns in $7$ hours.

  • The work rate for D&B Lawn Mowing's is, $\frac{5\text{ lawns}}{{6\frac{1}{2}}\text{ hours}}$. Divide by $2\frac{1}{2}$ to find D&B's unit work rate, $\frac{0.77\text{ lawns}}{1\text{ hour}}$.
  • The work rate for Richard's Landscape Service is, $\frac{7\text{ lawns}}{{8\frac{3}{4}}\text{ hours}}$. Divide by $8\frac{3}{4}$ to find Richard's unit work rate, $\frac{0.8\text{ lawns}}{1\text{ hour}}$.
  • The work rate for Sunrise Landscape is, $\frac{6\text{ lawns}}{7\text{ hours}}$. Divide by $7$ to find Sunrise's unit work rate, $\frac{0.86\text{ lawns}}{1\text{ hour}}$.
  • Since $\frac{0.86\text{ lawns}}{1\text{ hour}}>\frac{0.8\text{ lawns}}{1\text{ hour}}>\frac{0.77\text{ lawns}}{1\text{ hour}}$, Sunrise Landscape mows at a faster rate than the other mowing companies.

• Find the unit rate for each given rate.

  Hints

  A unit rate relates a quantity to one unit of another quantity.

  To find the unit rate, divide both sides of the fraction by the denominator.

  Rate: $\frac{84\text{ feet}}{12\text { seconds}}$

  Unit Rate: $\frac{7\text{ feet}}{1\text { second}}$

  Solution

  A unit work rate relates a quantity to one unit of another quantity.

  • The rate is $\frac{216\text{ feet}}{9\text { seconds}}$. To find the unit rate, divide the numerator and denominator by $9$: $\frac{24\text{ feet}}{1\text { second}}$.

  • The rate is $\frac{60\text{ feet}}{15\text { seconds}}$. To find the unit rate, divide the numerator and denominator by $15$: $\frac{4\text{ feet}}{1\text { second}}$.

  • The rate is $\frac{198\text{ feet}}{22\text { seconds}}$. To find the unit rate, divide the numerator and denominator by $22$: $\frac{9\text{ feet}}{1\text { second}}$.

  • The rate is $\frac{255\text{ feet}}{17\text { seconds}}$. To find the unit rate, divide the numerator and denominator by $17$: $\frac{15\text{ feet}}{1\text { second}}$.

  • The rate is $\frac{714\text{ feet}}{51\text { seconds}}$. To find the unit rate, divide the numerator and denominator by $51$: $\frac{14\text{ feet}}{1\text { second}}$.

  • The rate is $\frac{216\text{ feet}}{27\text { seconds}}$. To find the unit rate, divide the numerator and denominator by $27$: $\frac{8\text{ feet}}{1\text { second}}$.

• Identify and add work rates.

  Hints

  Writing work rates as fractions can help to visualize the work rate.

  The unit work rate is when the denominator of the fraction is $1$.

  Remember to convert all units to hours. There are $60$ minutes in $1$ hour.

  Solution

  First, write all work rates as fractions and convert minutes to hours by dividing the minutes by $60$:

  • Amy's work rate: $\frac{12\text{ pies}}{4\frac{1}{2}\text{ hours}}$
  • Frank's work rate: $\frac{8\text{ pies}}{177\text{ minutes}}=\frac{8\text{ pies}}{2.95\text{ hours}}$
  • Lisa's work rate: $\frac{15\text{ pies}}{296\text{ minutes}}=\frac{15\text{ pies}}{4.93\text{ hours}}$
  • Cece's work rate: $\frac{7\text{ pies}}{3\frac{1}{5}\text{ hours}}$
  • Juliet's work rate: $\frac{17\text{ pies}}{564\text{ minutes}}=\frac{17\text{ pies}}{9.4\text{ hours}}$

  Divide both the numerator and denominator by the denominator to find the unit work rates:

  • Amy's unit work rate: $\frac{2.67\text{ pies}}{1\text{ hour}}$
  • Frank's unit work rate: $\frac{2.71\text{ pies}}{1\text{ hour}}$
  • Lisa's unit work rate: $\frac{3.04\text{ pies}}{1\text{ hour}}$
  • Cece's unit work rate: $\frac{2.19\text{ pies}}{1\text{ hour}}$
  • Juliet's unit work rate: $\frac{1.81\text{ pies}}{1\text{ hour}}$

  Since the fractions all have a denominator of $1$ hour, you can answer questions $2-5$ by adding the numerators:

  • Amy's work rate is $2.67$ pies per hour.
  • If Amy and Frank work together, their unit work rate is $\frac{2.67\text{ pies}}{1\text{ hour}}+\frac{2.71\text{ pies}}{1\text{ hour}}=\frac{5.38\text{ pies}}{1\text{ hour}}$.
  • If Amy and Lisa work together, their unit work rate is $\frac{2.67\text{ pies}}{1\text{ hour}}+\frac{3.04\text{ pies}}{1\text{ hour}}=\frac{5.71\text{ pies}}{1\text{ hour}}$.
  • If Amy and Cece work together, their unit work rate is $\frac{2.67\text{ pies}}{1\text{ hour}}+\frac{2.19\text{ pies}}{1\text{ hour}}=\frac{4.86\text{ pies}}{1\text{ hour}}$.
  • If Amy and Juliet work together, their unit work rate is $\frac{2.67\text{ pies}}{1\text{ hour}}+\frac{1.81\text{ pies}}{1\text{ hour}}=\frac{4.48\text{ pies}}{1\text{ hour}}$.
  • Since $\frac{5.71\text{ pies}}{1\text{ hour}}>\frac{5.38\text{ pies}}{1\text{ hour}}>\frac{4.86\text{ pies}}{1\text{ hour}}>\frac{4.48\text{ pies}}{1\text{ hour}}$, the fastest to slowest partnership with Amy is Lisa, Frank, Cece, Juliet.