What are Proportional Relationships?

What are Proportional Relationships? exercise

• Describe how to set-up and solve proportional relationships.

  Hints

  The ratio of numbers $2$ and $3$ can be written in the following forms:

  • $2$ to $3$
  • $2:3$
  • $\frac{2}{3}$

  If it takes Flo $7$ minutes to run $1$ mile, how long does it take her to run $5$ miles?

  • The proportion is $\frac{7}{1}=\frac{t}{5}$, where $t$ represents time.
  • Cross multiply: $7(5)=1\cdot t$
  • Solve: $t=35$

  It will take Flo $35$ minutes to run $5$ miles.

  If it takes Aunt Sally $11$ minutes to run $1$ mile, how far will she be in $35$ minutes?

  • The proportion is $\frac{11}{1}=\frac{35}{d}$, where $d$ represents distance.
  • Cross multiply: $11d=35$
  • Solve: $t=3.18$

  Aunt Sally will be about $3.18$ miles in $35$ minutes.

  Solution

  A Proportional Relationship for Flo.

  Flo figures it will take $9$ minutes to run $1$ mile.

  This information can be written as a ratio because it tells us how much there is of one quantity when compared to another.

  The ratio as a fraction is, $\frac{9~\text{minutes}}{1~\text{mile}}$.

  Since the rate will never change, we can write a proportional relationship between time and distance.

  Let $t$ equal time, $\frac{9~\text{minutes}}{1~\text{mile}}=\frac{t}{3.1~\text{miles}}$.

  To solve for $t$, multiply the numerator on the left by the denominator on the right, and the numerator on the right by the denominator on the left.

  This is called cross multiplying, $9(3.1)=1\cdot t$.

  It will take Flo is approximately $28$ minutes to run the $5$k.

  A Proportional Relationship for Aunt Sally

  What about Aunt Sally? If it takes her $13$ minutes to run $1$ mile, where will Aunt Sally be in $28$ minutes, when Flo finishes the race?

  We can write another proportional relationship between time and distance.

  This time we want to solve for the distance which we can represent with the variable $d$.

  The proportion is represented by the equation,

  • $\frac{13}{1}=\frac{28}{d}$.

  Cross multiplying results in the equation,

  • $13d=28$.

  Solving the equation for $d$ means that Aunt Sally can go $2.15$ miles in $28$ minutes.

• Solve the proportion for the missing variable.

  Hints

  To solve a proportion, multiply the numerator on the left by the denominator on the right, and the numerator on the right by the denominator on the left. This is called cross multiplying.

  After cross multiplying a proportion, simplify the equation, then use the opposite operation, division, to solve for the variable.

  The following steps are used to solve the proportion, $\frac{t}{24}=\frac{9}{15}$:

  • Cross multiply: $24(9)=15t$
  • Simplify: $216=15t$
  • Solve: $t=14.4$

  Solution

  To solve a proportion, multiply the numerator on the left by the denominator on the right, and the numerator on the right by the denominator on the left. This is called cross multiplying.

  Simplify the equation, then use the opposite operation, division, to solve for the variable.

  1. $\frac{5}{t}=\frac{3}{25}$

  • Cross multiply: $5(25)=3t$
  • Simplify: $125=3t$
  • Solve: $t=41.7$

  2. $\frac{20}{7}=\frac{t}{16}$

  • Cross multiply: $20(16)=7t$
  • Simplify: $320=7t$
  • Solve: $t=45.7$

  3. $\frac{9}{t}=\frac{4}{19}$

  • Cross multiply: $9(19)=4t$
  • Simplify: $171=4t$
  • Solve: $t=42.8$

  4. $\frac{41}{87}=\frac{19}{t}$

  • Cross multiply: $41t=87(19)$
  • Simplify: $41t=1653$
  • Solve: $t=40.3$

  5. $\frac{t}{350}=\frac{8}{75}$

  • Cross multiply: $75t=350(8)$
  • Simplify: $75t=2800$
  • Solve: $t=37.3$

• Calculate and sort the proportional relationships.

  Hints

  The numbers in a ratio tell us how much there is of one quantity when compared to another.

  For example, $\frac{\text{miles}}{\text{hours}}$.

  To solve a proportion, multiply the numerator on the left by the denominator on the right, and the numerator on the right by the denominator on the left. This is called cross multiplying.

  Rita can read $19$ pages in $30$ minutes. How long will it take her to read a $250$ page book?

  • Set up the proportional relationship: $\frac{19}{30}=\frac{250}{t}$
  • Cross multiply: $19t=30(250)$
  • Simplify: $19t=7500$
  • Solve: $t=394.7$
  • It will take Rita about $395$ minutes to finish reading the book.

  Solution

  Write each word problem as a proportional relationship between two ratios, then solve for the unknown.

  It is always important to make sure that both numerators and denominators are consistent, $\frac{\text{miles}}{\text{hour}}=\frac{\text{miles}}{\text{hour}}$ or $\frac{\text{hours}}{\text{miles}}=\frac{\text{hours}}{\text{miles}}$.

  To solve a proportion, multiply the numerator on the left by the denominator on the right, and the numerator on the right by the denominator on the left. This is called cross multiplying.

  1. Ivanka is making cookies for her friends. The recipe calls for $2\frac{1}{4}$ cups of flour for $48$ servings. Ivanka wants to make $64$ servings. How many cups of flower does Ivanka need?

  • Let $c$ represent the cups of flour.
  • Set up the proportional relationship: $\frac{2\frac{1}{4}}{48}=\frac{c}{64}$
  • Cross multiply: $2\frac{1}{4}(64)=48c$
  • Simplify: $144=48c$
  • Solve: $c=3$
  • Ivanka needs $3$ cups of flour to make $64$ servings.

  2. Marty drove $10$ miles in $32$ minutes. If Marty keeps driving at the same rate, how many miles will he drive in $57$ minutes?

  • Let $m$ represent the number of miles.
  • Set up the proportional relationship: $\frac{10}{32}=\frac{m}{57}$
  • Cross multiply: $10(57)=32m$
  • Simplify: $570=32m$
  • Solve: $m=17.81$
  • Marty will drive $17.81$ miles in $57$ minutes.

  3. Rae knows that her $22$ pancakes can feed her family of $8$. How many pancakes are needed to feed $14$ people? Assume each person eats the same quantity of pancakes.

  • Let $p$ represent the number of pancakes.
  • Set up the proportional relationship: $\frac{22}{8}=\frac{p}{14}$
  • Cross multiply: $22(14)=8p$
  • Simplify: $308=8p$
  • Solve: $p=38.5$
  • Rae finds that $38.5$ pancakes will feed $14$ people.

  4. Sybille earns $15.25$ per hour at her job. How much will she earn after working $36$ hours?

  • Let $d$ represent the amount of money she will earn.
  • Set up the proportional relationship: $\frac{15.25}{1}=\frac{d}{36}$
  • Cross multiply: $15.25(36)=d$
  • Solve: $d=549$
  • Sybille will earn $549$ after working $36$ hours.

• Find the correct proportions, equations, and answer.

  Hints

  Make sure that both numerators and denominators in the proportional relationships are consistent.

  For example, $\frac{\text{miles}}{\text{hour}}=\frac{\text{miles}}{\text{hour}}$ or $\frac{\text{hours}}{\text{miles}}=\frac{\text{hours}}{\text{miles}}$.

  To solve a proportion, cross multiply the numerators and denominators.

  Johann can run $3$ miles in $32$ minutes. How long will it take him to run $8.5$ miles?

  • Set up the proportional relationship: $\frac{3}{32}=\frac{8.5}{t}$
  • Cross multiply: $3t=32(8.5)$
  • Simplify: $3t=272$
  • Solve: $t=90.7$
  • It will take Johann about $91$ minutes to run $8.5$ miles.

  Solution

  Anna sells homemade cherry pies at a weekly bake sale. Last Sunday she sold $14$ pies and made $ \$108.50$. If she makes $ \$209.25$ the following week, how many pies did she sell?

  Proportions

  Since Anna sells $14$ pies for $ \$108.50$, the ratio is, $\frac{14}{108.50}$.

  Anna wants to know how many pies she needs to sell to make $ \$209.25$ so she sets up the proportion, $\frac{14}{108.50}=\frac{x}{209.25}$, where $x$ represents the number of pies.

  It is always important to make sure that both numerators and denominators are consistent, $\frac{\text{pies}}{\text{total sales}}=\frac{\text{pies}}{\text{total sales}}$ or $\frac{\text{total sales}}{\text{pies}}=\frac{\text{total sales}}{\text{pies}}$.

  This means that another way to write the proportion is, $\frac{108.50}{14}=\frac{209.25}{x}$.

  Equations

  To solve a proportion, cross multiply the numerator on the left by the denominator on the right, and the denominator on the left by the numerator on the right.

  • 14(209.25)=108.50x$

  Simplify,

  • $108.50x=2,929.50$

  Answer

  To solve the proportion, use opposite operations to isolate the variable $x$.

  • Divide both sides by $108.50$,
  • $x=27$.

  Therefore, Anna has to sell 27 pies to make $ \$209.25$.

• Identify which pairs of ratios are proportional to each other and which are not.

  Hints

  Two ratios are proportional to each other if they are equivalent ratios.

  $\frac{18}{3}$ and $\frac{6}{1}$ are proportional because they both reduce to the fraction $\frac{6}{1}$.

  $\frac{8}{12}$ and $\frac{3}{4}$ are not proportional to each other because they do not reduce to the same fraction, so they are not equivalent ratios.

  Solution

  Proportional

  The following pairs of ratios are proportional to each other because both reduce to the same fraction, meaning that they are equivalent ratios:

  • $\frac{4}{8}$ and $\frac{5}{10}$ simplifies to $\frac{1}{2}$ and $\frac{1}{2}$.

  • $\frac{2}{8}$ and $\frac{4}{16}$ simplifies to $\frac{1}{4}$ and $\frac{1}{4}$.

  • $\frac{10}{5}$ and $\frac{16}{8}$ simplifies to $\frac{2}{1}$ and $\frac{2}{1}$.

  • $\frac{18}{4}$ and $\frac{9}{2}$ simplifies to $\frac{9}{2}$ and $\frac{9}{2}$.

  Not Proportional

  The following relationships are not proportional because simplifying the original ratios results in non-equivalent ratios:

  • $\frac{5}{12}$ and $\frac{12}{3}$ simplifies to $\frac{5}{12}$ and $\frac{4}{1}$.

  • $\frac{3}{9}$ and $\frac{4}{10}$ simplifies to $\frac{1}{3}$ and $\frac{2}{5}$.

  • $\frac{15}{5}$ and $\frac{5}{15}$ simplifies to $\frac{3}{1}$ and $\frac{1}{3}$.

  • $\frac{1}{9}$ and $\frac{27}{3}$ simplifies to $\frac{1}{9}$ and $\frac{9}{1}$.

• Solve the real-world proportion problems.

  Hints

  It may be necessary to convert units to make them the same.

  For example:

  • $4$ feet $3$ inches is $15$ inches
  • $3$ hours and $20$ minutes is $200$ minutes.

  Make sure that both numerators and denominators in the proportional relationships are consistent.

  For example, $\frac{\text{miles}}{\text{hour}}=\frac{\text{miles}}{\text{hour}}$ or $\frac{\text{hours}}{\text{miles}}=\frac{\text{hours}}{\text{miles}}$.

  Rita can sew $3$ blankets in $5\frac{1}{2}$ days. How many days will it take Rita to sew $13$ blankets?

  • Set up the proportional relationship: $\frac{3}{5\frac{1}{2}}=\frac{13}{d}$
  • Cross multiply: $5\frac{1}{2}(13)=3d$
  • Simplify: $71\frac{1}{2}=3d$
  • Solve: $d=23.83$
  • It will take Johann about $23.8$ days to make $13$ blankets.

  Solution

  1. Dr. Cheng prides himself on seeing many patients in one day. On a normal day, he evaluates $3$ patients in $1$ hour and $15$ minutes. How many patients will Dr. Cheng see in $6$ hours and $40$ minutes?

  • Let $p$ represent the number of patients.
  • Convert hours to minutes: $1$ hour and $15$ minutes is $75$ minutes, and $6$ hours and $40$ minutes is $400$ minutes.
  • Set up the proportional relationship: $\frac{3}{75}=\frac{p}{400}$
  • Cross multiply: $3(400)=75p$
  • Simplify: $1,200=75p$
  • Solve: $p=16$
  • Dr. Cheng will see $16$ patients in $6$ hours and $40$ minutes.

  2. Three friends go on a camping trip every year. Last year they made sure to have $9$ bottles of water every day. If $5$ friends are going on the camping trip this year, how many bottles of water do they need to bring for each day?

  • Let $w$ represent the number of water bottles.
  • Set up the proportional relationship: $\frac{3}{9}=\frac{5}{w}$
  • Cross multiply: $3w=9(5)$
  • Simplify: $3w=45$
  • Solve: $w=15$
  • The $5$ friends need to bring $15$ water bottles.

  3. Aaron has a large baseball card collection. He has been collecting cards since he was a little boy. The baseball cards come in packs of $12$. If Aaron has $540$ cards, how many packs did he buy?

  • Let $p$ represent the packs of baseball cards.
  • Set up the proportional relationship: $\frac{12}{1}=\frac{540}{p}$
  • Cross multiply: $12p=1(540)$
  • Simplify: $12p=540$
  • Solve: $p=45$
  • Aaron bought $45$ packs of baseball cards.

  4. Sarita builds model cars as a hobby. She is very cautious about making all of her cars to scale with the real car. Sarita is building a model of a sports car that is $10\frac{3}{4}$ feet long and $4\frac{3}{4}$ feet high. If she makes the model car $5$ inches long, what will be the height of the model car be?

  • Let $h$ represent the height of the model car.
  • Convert feet to inches: $10\frac{3}{4}\text {feet}=129\text {inches}$ and $4\frac{1}{2}\text {feet}=54\text {inches}$
  • Set up the proportional relationship: $\frac{129}{54}=\frac{5}{h}$
  • Cross multiply: $129h=54(5)$
  • Simplify: $129h=270$
  • Solve: $h=2.09$
  • Sarita needs to make her model car $2.1$ inches high.