Equivalent Ratios Given Parts and Wholes

Equivalent Ratios Given Parts and Wholes exercise

• Given a ratio, find an equivalent ratio.

  Hints

  Start by setting up a ratio of $a$ to $b$. This represents the constant of proportionality, $k$.

  The ratio of workout days to rest days is $\frac{5}{1}$.

  Solution

  According to Mr. Bull, the best workout plan in a month with 30 total days, there should be 25 workout days, and 5 rest days.

  $\frac{25}{5}=\frac{5}{1}$

  $25+5=30$

• Apply the process needed to complete the ratio table.

  Hints

  The unknown variable in an equation is the variable that you are solving for.

  Always simplify the constant of proportionality.

  If the constant of proportionality is $\frac{350}{150}$, simplify by dividing both numbers by the greatest common factor. In this case, the GCF is $50$ and $\frac{350\div 50}{150\div 50}=\frac{7}{3}$.

  Solution

  Use the values in the table to create a ratio representing Mr. Bull's leg and arm workout: $k=\frac{300}{200}$ and then simplify the ratio to: $\bf{\frac{3}{2}}$

  Using the variables $L$ for legs and $A$ for arms, we can then identify the equation: $\frac{L}{A}=\frac{3}{2}$.

  For Scarlett's workout plan, Mr. Bull already knows that he wants her leg workout to include $4\frac{1}{2}$ lbs, which can be written as an improper number: $\bf{\frac{9}{2}}$

  Since Mr. Bull wants to make her workout proportional to his, he will use the same equation, $\frac{L}{A}=\frac{3}{2}$, and the unknown variable is $\bf{A}$ (arms).

  $\dfrac{\frac{9}{2}}{A}=\dfrac{3}{2}$

  Cross multiplication can be used to find the missing variable of $A$.

  $\begin{array}{l}\frac{\frac{9}{2}}{A}=\frac{3}{2}\\ \\ 3\left(A\right)=\frac{9}{2}\left(2\right)\\ \\ \frac{3A}{3}=\frac{9}{3}\end{array}$

  $A$=$\bf{3}$ lbs

  Scarlett's total workout is:

  $L+A=4\frac{1}{2}+3$ = $\bf{7\frac{1}{2}}$ lbs

• Find the proportional equation and use it to solve.

  Hints

  The word problem represents a proportional equation where the constant of proportionality, $k$, is equal to the ratio of water to pots, $\frac{W}{P}$, or pots to water, $\frac{P}{W}$.

  To find the constant of proportionality, $k$, simplify the fraction of $\frac{W}{P}$ or $\frac{P}{W}$.

  To solve a proportional equation, use opposite operations or cross multiply.

  There are a total of three true statements.

  Solution

  The word problem represents a proportional equation where the constant of proportionality, $k$, is equal to the ratio of water to pots, $\frac{W}{P}$, or pots to water, $\frac{P}{W}$.

  • $\frac{W}{P}=k$ or $\frac{P}{W}=k$

  To find the constant of proportionality, $k$, substitute $32$ for $P$ and $8$ for $W$.

  • $\frac{W}{P}=\frac{8}{32}$ or $\frac{P}{W}=\frac{32}{8}$

  Simplify:

  • $\frac{W}{P}=\frac{1}{4}$ or $\frac{P}{W}=\frac{4}{1}$

  To find how many pots Rosie can water, substitute $23$ for $W$ into one of the equations.

  • $\frac{23}{P}=\frac{1}{4}$

  Cross multiply and solve:

  • $P\cdot 1=23\cdot 4$
  • $P=92$

  Therefore, Rosie can water $92$ flower pots with $23$ liters of water.

• Understand how a ratio table works.

  Hints

  Write an equation by setting the constant of proportionality, $k$, equal to the ratio of bananas to strawberries, $\frac{B}{S}$.

  When solving for an unknown, cross multiply or use opposite operations.

  When adding fractions, make sure both fractions have a common denominator, then add the numerators.

  • Example: $2\frac{3}{7}+4\frac{5}{7}=6\frac{8}{7}=7\frac{1}{7}$

  Solution

  Mr. Bull

  • Use the given information to fill in the first two columns in the table.
  • Add bananas and strawberries together to find the total: $4+12=16$
  • Find the constant of proportionality, $k$, and write an equation using the ratio of bananas to strawberries.
  • $k=\frac{4}{12}=\frac{1}{3}$
  • Since $k=\frac{B}{S}$, the equation is $\frac{B}{S}=\frac{1}{3}$.

  Rachel

  • Since we know that there are $3\frac{1}{4}$ bananas in Rachel's smoothie, we can substitute $B$ into the equation.
  • First, convert $3\frac{1}{4}$ into an improper fraction, $\frac{13}{4}$.
  • $\frac{\frac{13}{4}}{S}=\frac{1}{3}$
  • Cross multiply to solve for $S$.
  • $S\cdot 1=\frac{13}{4}\cdot \frac{3}{1}$
  • $S=\frac{39}{4}=9\frac{3}{4}$
  • To find the total add bananas and strawberries: $3\frac{1}{4}+9\frac{3}{4}=12\frac{4}{4}=13$

  Trisha

  • Convert $5\frac{1}{4}$ into an improper fraction, $\frac{21}{4}$, then substitute for $S$ in the equation, $\frac{B}{S}=\frac{1}{3}$.
  • $\frac{B}{\frac{21}{4}}=\frac{1}{3}$
  • Use opposite operations and multiply both sides by $\frac{21}{4}$.
  • $B=\frac{1}{3}\cdot \frac{21}{4}=\frac{21}{12}=1\frac{9}{12}=1\frac{3}{4}$
  • To find the total, add bananas and strawberries: $1\frac{3}{4}+5\frac{1}{4}=6\frac{4}{4}=7$

  Marvin

  • Since we only know the total, create a new equation using a new ratio, $\frac{\text{Strawberries}}{\text{Total}}$.
  • We can use any of the numbers in the table, but the easiest ratio can be found from using Mr. Bull's numbers: $\frac{S}{T}=\frac{12}{16}$
  • The new equation is, $\frac{S}{T}=\frac{3}{4}$.
  • Convert Martin's total, $6\frac{2}{5}$, into an improper fraction, $\frac{32}{5}$, and substitute for $T$ in the equation.
  • $\frac{S}{\frac{32}{5}}=\frac{3}{4}$
  • Use opposite operations and multiply both sides by $\frac{32}{5}$ to isolate $S$.
  • $S=\frac{3}{4}\cdot \frac{32}{5}=\frac{96}{20}=\frac{24}{5}=4\frac{4}{5}$
  • To find the number of bananas, subtract strawberries from the total: $6\frac{2}{5}-4\frac{4}{5}=5\frac{7}{5}-4\frac{4}{5}=1\frac{3}{5}$

• Understanding equivalent ratios to find a ratio in simplest form.

  Hints

  Remember, a ratio compares the values of two different quantities. For example if you were making lemonade, you may use a ratio comparing the number of lemons to the cups of sugar.

  Example: 8 lemons for every 1 cup of sugar

  Ratios can be written in three different ways:

  • $8$ to $1$
  • $\frac{8}{1}$
  • $8:1$

  Equivalent ratios can be found by multiplying or dividing a ratio by a number to scale it up or scale it down.

  For example, if we were to follow the recipe of $8$ lemons for every $1$ cup of sugar, but we wanted more lemonade, we can multiply the numbers by $3$ to find an equivalent ratio of $\frac{24}{3}$.

  If we had a ratio of $\frac{10}{5}$ we can divide both the numerator and denominator by $5$ to simplify the ratio to it's lowest terms.

  $\frac{10}{5}=\frac{2}{1}$

  Solution

  For every 2 bananas, there is 1 tablespoon of peanut butter.

  The ratios in the table can all be simplified to $\frac{2}{1}$.

• Decide which model represents the ratio table.

  Hints

  Find the constant of proportionality, $k$, by setting up a ratio and simplifying.

  The shaded portion in tape diagrams and circle models represents the part, and the entire diagram represents the whole.

  In a double number line, the top line represents the part and the bottom line represents the whole.

  Solution

  Table 1: $\begin{array}{|c|c|} \hline \\ 3\frac{1}{3} & \frac{20}{21}\\ \\ \hline \\ 4 & \frac{8}{7}\\ \\ \hline \\ 4\frac{1}{2} & 1\frac{2}{7}\\ \\ \hline \end{array}$

  • To find the constant of proportionality, $k$, set up a ratio between two numbers in any row of the ratio table.
  • Using the first row of numbers, the constant of proportionality is, $k=\frac{3\frac{1}{3}}{\frac{20}{21}}=\frac{\frac{10}{3}}{\frac{20}{21}}=\frac{10}{3}\cdot \frac{21}{20}=\frac{210}{60}=\frac{7}{2}$
  • The visual representation that shows the constant of proportionality, $k=\frac{7}{2}$, is the tape diagram.
  • This is because $\frac{7}{2}$ written as a mixed number is $3\frac{1}{2}$, and $3$ and$\frac{1}{2}$ boxes are shaded in.

  Table 2: $\begin{array}{|c|c|} \hline \\ \frac{3}{4} & \frac{3}{2}\\ \\ \hline \\ 1\frac{1}{2} & 3\\ \\ \hline \\ 2\frac{1}{5} & 4\frac{2}{5}\\ \\ \hline \end{array}$

  • To find the constant of proportionality, $k$, set up a ratio between two numbers in any row of the ratio table.
  • Using the second row of numbers, the constant of proportionality is, $k=\frac{1\frac{1}{2}}{3}=\frac{\frac{3}{2}}{3}=\frac{3}{2}\cdot \frac{1}{3}=\frac{3}{6}=\frac{1}{2}$.
  • The visual representation that shows the constant of proportionality, $k=\frac{1}{2}$, is the double number line
  • This is because the last number on the top line, $50$, is the part when the whole is $100$, and $\frac{50}{100}=\frac{1}{2}$.

  Table 3: $\begin{array}{|c|c|} \hline \\ 2 & 1\frac{3}{5}\\ \\ \hline \\ 2\frac{1}{4} & 1\frac{4}{5}\\ \\ \hline \\ 2\frac{3}{4} & 2\frac{1}{5}\\ \\ \hline \end{array}$

  • To find the constant of proportionality, $k$, set up a ratio between two numbers in any row of the ratio table.
  • Using the third row of numbers, the constant of proportionality is $k=\frac{2\frac{3}{4}}{2\frac{1}{5}}=\frac{\frac{11}{4}}{\frac{11}{5}}=\frac{11}{4}\cdot \frac{5}{11}=\frac{55}{44}=\frac{5}{4}$
  • The visual representation that shows the constant of proportionality, $k=\frac{5}{4}$, is the fraction $\frac{15}{12}$.
  • This is because dividing both numbers in the fraction, $\frac{15}{12}$, by the greatest common factor of $3$ yields $\frac{5}{4}$.

  Table 4: $\begin{array}{|c|c|} \hline \\ 2\frac{1}{3} & 7\\ \\ \hline \\ 4\frac{1}{2} & 9\\ \\ \hline \\ 5 & 15\\ \\ \hline \end{array}$

  • To find the constant of proportionality, $k$, set up a ratio between two numbers in any row of the ratio table.
  • Using the first row of numbers, the constant of proportionality is $k=\frac{2\frac{1}{3}}{7}=\frac{\frac{7}{3}}{7}=\frac{7}{3}\cdot \frac{1}{7}=\frac{7}{21}=\frac{1}{3}$
  • The visual representation that shows the constant of proportionality, $k=\frac{1}{3}$, is the circle model.
  • This is because $2$ of the $6$ pieces are shaded which yields the fraction $\frac{2}{6}=\frac{1}{3}$.