Ratio and Proportions

Ratio and Proportions exercise

• Explain the ratio of the ingredients for Dr. Evil's smoothie.

  Hints

  Imagine the monstera deliciosa being on one side and the Brussels sprouts on the other side.

  Now write the number of parts below each ingredient and put a $:$ sign in between.

  To reduce a fraction, we divide the numerator and the denominator by the same number.

  Take a look at the following example:

  $\frac8{12}=\frac{8\div 4}{12\div 4}=\frac23$.

  Solution

  We know that Dr. Evil needs six parts of monstera deliciosa and only two parts of Brussels sprouts for his smoothie. This can be written as following:

  $\begin{array}{ccc} \text{monstera deliciosa }&&\text{Brussels sprouts}\\ 6&:&2 \end{array}$

  This is the ratio, and it can also be written in fraction form:

  $\frac62=\frac{6\div2}{2\div2}=\frac31$.

  Now we change the order to the ratio of Brussels sprouts to monstera deliciosa.

  $\begin{array}{ccc} \text{brussels sprouts}&&\text{monstera deliciosa }\\ 2&:&6 \end{array}$

  This is the ratio. Written in fraction form:

  $\frac26=\frac{2\div2}{6\div2}=\frac13$.

• Determine how many women are interested in Dr. Evil.

  Hints

  The ratio of women interested in Dr. Evil compared to all women is equal to $2:8$.

  Write $2:8$ as a fraction then reduce it to lowest terms.

  If the fraction is given by $\frac ?4$ you can find the answer in the numerator.

  Solution

  Dr. Evil knows that two out of eight women will be interested in him. This can be written as $2:8$.

  To calculate other combinations, we can write this as a equivalent fraction and fill in the unknown values:

  $\frac28=\frac{2\div 2}{8\div 2}=\frac14$.

  Now we also know that one out of four women will be interested in Dr. Evil.

  So if he meets $4$ women, $1$ of them will be interested in him.

• Determine the right ratio.

  Hints

  Reduce each fraction to its lowest terms.

  You can reduce a fraction by dividing the numerator and the denominator by the same number.

  Take a look at the following example:

  $\frac{121}{22}=\frac{121\div11}{22\div11}=\frac{11}2$

  Solution

  To decide which fraction belongs to which ratio, we have to reduce each fraction to its lowest terms:

  • $\frac{10}{15}=\frac{10\div5}{15\div5}=\frac23$ $\rightarrow 2:3$
  • $\frac{22}{33}=\frac{22\div11}{33\div11}=\frac23$ $\rightarrow 2:3$
  • $\frac{4}{6}=\frac{4\div2}{6\div2}=\frac23$ $\rightarrow 2:3$
  • $\frac{50}{60}=\frac{50\div10}{60\div10}=\frac56$ $\rightarrow 5:6$
  • $\frac{15}{18}=\frac{15\div3}{18\div3}=\frac56$ $\rightarrow 5:6$
  • $\frac{49}{63}=\frac{49\div7}{63\div7}=\frac79$ $\rightarrow 7:9$
  • $\frac{21}{27}=\frac{21\div3}{27\div3}=\frac79$ $\rightarrow 7:9$

• Determine the corresponding ratios.

  Hints

  Be careful with the order of the ratio.

  Write the given ratio as a fraction and reduce the fraction as much as possible.

  Solution

  For each relation we write the ratio as a fraction and reduce this fraction to its lowest terms.

  • Dragon blood to tear drops: $\frac5{15}=\frac{5\div5}{15\div 5}=\frac13~\rightarrow~1:3$
  • Snake poison to water: $\frac{20}{35}=\frac{20\div5}{35\div 5}=\frac47~\rightarrow~4:7$
  • Water to dragon blood: $\frac{35}{5}=\frac{35\div5}{5\div 5}=\frac71~\rightarrow~7:1$
  • Tear drops to snake poison: $\frac{15}{20}=\frac{15\div5}{20\div 5}=\frac34~\rightarrow~3:4$
  • Water to snake poison: $\frac{35}{20}=\frac{35\div5}{20\div 5}=\frac74~\rightarrow~7:4$
  • Snake poison to tear drops: $\frac{20}{15}=\frac{20\div5}{15\div 5}=\frac43~\rightarrow~4:3$

• Describe the difference between ratio and proportion.

  Hints

  $6:2$ and $3:1$ are both ratios.

  A proportion is like an equation.

  Solution

  Keep in mind:

  Ratios are used to compare quantities.

  Two equal ratios create a proportion.

  For example:

  • The ratio is $6:2$ or $3:1$.
  • The proportion is $\frac 62 = \frac 31$.

• Calculate the number of ingredients for 30 stinging nettle plants.

  Hints

  The ratio stinging nettle to cod liver oil is $3:1$.

  Write this ratio as a fraction then write an equivalent fraction with a numerator equal to $30$.

  Then you can solve for the unknown denominator.

  The ingredients in increasing order:

  1. cod liver oil
  2. stinging nettle
  3. chamomile
  4. Artemisia

  The total number of cod liver oil, stinging nettle, and chamomile plants is 100.

  Solution

  We know that the number of stinging nettle plants is $30$.

  Let's begin with cod liver oil: The number of stinging nettle plants is three times the number of the cod liver oil. That means $3:1$. We write this ratio as a fraction and raise the terms by $10$:

  $\frac 31=\frac{3\times 10}{1\times 10}=\frac{30}{10}$.

  • In the numerator we have the number of stinging nettle plants: $30$.
  • The denominator shows the number of the cod liver oil: $10$.

  Now we take a look at the chamomile plants: The number of stinging nettle plants is only half the chamomile plants. Here we have $1:2$.

  We can write this as a fraction $\frac12$ and raise the terms by $30$:

  $\frac 12=\frac{1\times 30}{2\times 30}=\frac{30}{60}$.

  • In the numerator we have the number of stinging nettle plants: $30$
  • The denominator shows the number of the chamomile plants: $60$

  In total we have $30+10+60=100$ different plants.

  Finally, we calculate the number of Artemisia plants: the number of Artemisia plants must always be twice as much as the number of all other ingredients. This is given by $2:1$. Let's write it as a fraction and raise the terms by $100$, the total number of different plants:

  $\frac 21=\frac{2\times 100}{1\times 100}=\frac{200}{100}$.

  • In the denominator we have the total number of all other plants: $100$.
  • The numerator shows the number of artemisia plants: $200$.