Associated Ratios and the Value of a Ratio

Associated Ratios and the Value of a Ratio exercise

• Find the associated ratio for each given ratio.

  Hints

  The ratio $1$ to $2$ can be written as $1:2$ or $\frac{1}{2}$.

  The numbers in a ratio can be used to form different ratios, which are called associated ratios.

  Two associated ratios for $\frac{1}{4}$ are $4:1$ and $4:3$.

  Solution

  Associated ratios are related to one another. This means that given one ratio, we can find the other.

  $5$ to $1$ is associated with $\frac{1}{5}$

  • The numbers in the ratio are switched.

  $4:3$ is associated with $\frac{9}{12}$

  • Simplify $\frac{9}{12}$ by dividing both numbers by $3$ yields $\frac{3}{4}$.
  • $\frac{3}{4}$ is associated with $4:3$.

  $1$ to $7$ is associated with $6:7$

  • Since $7-1=6$, $6:7$ is associated with $1$ to $7$.

  $\frac{18}{4}$ is associated with $2$ to $9$

  • Simplify $\frac{18}{4}$ by dividing both numbers by $2$ yields $\frac{9}{2}$.
  • $\frac{9}{2}$ is associated with $\frac{18}{4}$.

  $6:10$ is associated with $5$ to $3$

  • Simplify $6:10$ by dividing both numbers by $2$ yields $3:5$.
  • $3:5$ is associated with $6:10$.

  $\frac{1}{2}$ is associated with $11:22$

  • Simplify $11:22$ by dividing both side by $11$ yields $1:2$.
  • $1:2$ is associated with $\frac{1}{2}$.

• Write each sentence in ratio notation.

  Hints

  Equivalent ratios can be found either by simplifying both sides using the GCF, or by multiplying both sides by a number.

  There are two string instruments for every four percussion instruments.

  • The ratio is $2:4$ or $\frac{2}{4}$
  • If you divide both numbers by $2$, the ratio becomes $1:2$ or $\frac{1}{2}$.

  You can also multiply the ratio $1:2$ or $\frac{1}{2}$ by a number to find an equivalent ratio.

  • If you multiply both numbers by $3$, the ratio becomes $3:6$ or $\frac{3}{6}$.

  Solution

  1. There are five string instruments for every three brass instruments.

  • The ratio is written as $5:3$ or $\frac{5}{3}$, which equals $\frac{10}{6}$.
  • If you multiply both numbers in the ratio $\frac{5}{3}$ by $2$, the ratio becomes $\frac{10}{6}$.

  2. There are four woodwinds for every one percussion instrument.

  • The ratio is written as $4:1$ or $\frac{4}{1}$, which equals $\frac{12}{3}$.
  • If you multiply both numbers in the ratio $\frac{4}{1}$ by $3$, the ratio becomes $\frac{12}{3}$.

  3. There are three string instruments for every two woodwind instruments.

  • The ratio is written as $3:2$ or $\frac{3}{2}$.
  • If you multiply both numbers in the ratio $3:2$ by $2$, the ratio becomes $6:4$.
  • If you multiply both numbers in the ratio $\frac{3}{2}$ by $3$, the ratio becomes $\frac{12}{8}$.

  4. There are two brass instruments for every five woodwind instruments.

  • The ratio is written as $2:5$ or $\frac{2}{5}$, which equals $\frac{6}{15}$.
  • If you multiply both numbers in the ratio $\frac{2}{5}$ by $3$, the ratio becomes $\frac{6}{15}$.

  5. There are two percussion instruments for every eight string instruments.

  • The ratio is written as $2:8$ or $\frac{2}{8}$.
  • If you divide both numbers by two, the ratios become $1:4$ and $\frac{1}{4}$.
  • If you multiply both numbers in the ratio $\frac{1}{4}$ by $3$, the ratio becomes $\frac{3}{12}$.

  6. There are six brass instruments for every three percussion instruments.

  • The ratio is written as $6:3$ or $\frac{6}{3}$.
  • If you divide both numbers by $3$, the ratios become $2:1$ and $\frac{2}{1}$.
  • If you multiply both numbers in the ratio $\frac{2}{1}$ by $2$, the ratio becomes $\frac{4}{2}$.

• Identify which statements are true.

  Hints

  $4:1$, $\frac{4}{1}$, and $4$ to $1$ are three ways of representing a ratio.

  If there are three apples to two bananas, the ratio can be written as $3:2$.

  If there are three apples to two bananas, the associated ratio can be written as $2:3$.

  Solution

  True Statements:

  An associated ratio can be formed from the numbers of a ratio.

  • For example, the numbers in the ratio $3:2$ can be switched to write the associated ratio $2:3$.

  The ratio $4:8$ is equal to $2:4$.

  • If you divide both numbers by $2$, the ratio $4:8$ equals $2:4$.

  A ratio compares values.

  • If there are $5$ girls for every $3$ boys, the ratio $5$ to $3$ compares the values of girls to boys.

  An associated ratio for $\frac{4}{10}$ is $\frac{5}{2}$.

  • If you divide both numbers by $2$, $\frac{4}{10}$ equals $\frac{2}{5}$. Switching the numbers in the ratio yields the associated ratio $\frac{5}{2}$.

  An associated ratio for $5:7$ is $2:7$.

  • Since $7-5=2$, the ratio $2:7$ is an associated ratio for $5:7$.

  False Statements:

  The ratio $3:8$ is equal to $8:3$.

  • $8:3$ is an associated ratio to $3:8$ but they are not equal. This is because order matters in a ratio. Comparing the values $3$ to $8$ is very different from comparing the values of $8$ to $3$.

  Associated ratios are always equal.

  • Associated ratios are not always equal because they can be formed by rearranging numbers in a ratio. For example, $\frac{1}{9}$ is not equal to the associated ratio $\frac{9}{1}$.

  The ratio $\frac{2}{10}$ is equal to $5:1$

  • If you divide both numbers by $2$, $\frac{2}{10}$ equals $\frac{1}{5}$, which does not equal the associated ratio, $\frac{5}{1}$.

• Identify the ratio which is being described along with its associated ratio.

  Hints

  A ratio is a relation between two quantities.

  An associated ratio can be formed from the quantities of another ratio.

  An artist uses four yellow tubes of paint for every one red tube of paint.

  • The ratio for yellow tubes to red tubes is $4:1$ or $\frac{4}{1}$.
  • The associated ratio for yellow tubes to red tubes is $1:4$ or $\frac{1}{4}$.

  Solution

  A ratio is a relation between two amounts.

  An associated ratio can be formed from the quantities of another ratio.

  1. Rachel went to the grocery store and bought five bananas for every two apples.

  • The ratio of bananas to apples is $5:2$ or $\frac{5}{2}$.
  • The associated ratio of apples to bananas is $2:5$ or $\frac{2}{5}$.

  2. Jaquan is making a smoothie and puts in three strawberries for every four blueberries.

  • The ratio of strawberries to blueberries is $3:4$ or $\frac{3}{4}$.
  • The associated ratio of blueberries to strawberries is $4:3$ or $\frac{4}{3}$.

  3. In Lingda's classroom there are six boys for every five girls.

  • The ratio of boys to girls is $6:5$ or $\frac{6}{5}$.
  • The associated ratio of girls to boys is $5:6$ or $\frac{5}{6}$.

  4. In the month of May, there were three sunny days for every one rainy day.

  • The ratio of sunny days to rainy days is $3:1$ or $\frac{3}{1}$.
  • The associated ratio of rainy days to sunny days is $1:3$ or $\frac{1}{3}$.

  5. A farmer planted seven sunflowers every three feet.

  • The ratio of sunflowers to feet is $7:3$ or $\frac{7}{3}$.
  • The associated ratio of feet to sunflowers is $3:7$ or $\frac{3}{7}$.

• Write each fraction in ratio notation.

  Hints

  A ratio written in fraction notation is the same as a ratio written in ratio notation.

  Both ways of representing the ratio, $1:2$ and $\frac{1}{2}$, are read "one to two".

  The ratio notation for $\frac{7}{3}$ is $7:3$.

  Solution

  A ratio written in fraction notation is the same as a ratio written in ratio notation.

  The numerator always corresponds to the left side of a ratio, and the denominator always corresponds to the right side of a ratio.

  1. The ratio notation for $\frac{3}{2}$ is $3:2$.

  2. The ratio notation for $\frac{7}{3}$ is $7:3$.

  3. The ratio notation for $\frac{1}{9}$ is $1:9$.

  4. The ratio notation for $\frac{11}{7}$ is $11:7$.

  5. The ratio notation for $\frac{5}{19}$ is $5:19$.

  6. The ratio notation for $\frac{8}{3}$ is $8:3$.

• Determine the ratio and its associated ratio.

  Hints

  A ratio is a relation between two amounts.

  An associated ratio can be formed from the quantities of another ratio.

  Remember to always simplify your ratios.

  A farmer owns ten cows, eighteen chickens and seven goats.

  • The ratio of cows to chicken and goats is $10:25$ or $\frac{10}{25}$ which simplifies to $2:5$ or $\frac{2}{5}$.
  • The associated ratios of cows to all of the animals is $10:36$ or $\frac{10}{36}$ which simplifies to $5:18$ or $\frac{5}{18}$.

  Solution

  A ratio is a relation between two amounts, and an associated ratio can be formed from the quantities of another ratio.

  Remember to always simplify your ratios and that order matters!

  1. Johann is making cookies. The recipe requires two eggs, three cups of flour, and one cup of sugar.

  • The ratio of eggs to the total number of ingredients is $2:6$ or $\frac{2}{6}$, which simplifies to $1:3$ or $\frac{1}{3}$.
  • The associated ratio of the total number of ingredients to everything but eggs is $6:4$ or $\frac{6}{4}$, which simplifies to $3:2$ or $\frac{3}{2}$.

  2. An orchestra is practicing for opening night. There are five flutes, five violins, three clarinets, and one piano.

  • The ratio of pianos and clarinets to the remaining instruments is $4:10$ or $\frac{4}{10}$, which simplifies to $2:5$ or $\frac{2}{5}$.
  • The associated ratio of flutes and violins to the remaining instruments is $10:4$ or $\frac{10}{4}$, which simplifies to $5:2$ or $\frac{5}{2}$.

  3. Hazel is making a list for her birthday party. She is inviting fifteen girls and eight boys.

  • The ratio of boys to girls is $8:15$ or $\frac{8}{15}$.
  • The associated ratio of girls to the total number of friends coming to the party is $15:23$ or $\frac{15}{23}$.

  4. There are nine boys and seven girls in math class. The students are surveyed about their favorite sports. The results are: $4$ Basketball, $6$ Soccer, $3$ Tennis, and $3$ Other.

  • The ratio of basketball to soccer and tennis is $4:9$ or $\frac{4}{9}$.
  • The associated ratio of basketball to all three sports is $4:13$ or $\frac{4}{13}$.