Solving Proportions

Solving Proportions exercise

• Manipulate the formula to get the value for $f$.

  Hints

  Here you can see an example of cross-multiplication:

  $\frac42=\frac a3~\leftrightarrow~4\times 3=2\times a$

  You can solve an equation by using opposite operations:

  • the opposite operation of addition is subtraction and vice versa
  • the opposite operation of multiplication is division and vice versa

  Solution

  An equation representing a proportion has the form: $\frac ab=\frac cd$.

  You have to know three values to get the solution for the unknown value. First of all, you cross-multiply the terms: $a\times d=c\times b$.

  Afterwards you can solve this equation by dividing.

  Now, let's have a look at the equation:

  $\frac1{0.5}=\frac f2$

  • first cross-multiply to $1\times 2=0.5\times f$
  • divide by $0.5$ to get the solution, $f=4$

• Determine the depth of the snow after two hours.

  Hints

  In order to solve equations you need to use opposite operations.

  The equation $2x=6$ can be solved by dividing by $2$.

  A ratio like four cookies for two yetis can be written as $\frac42$.

  Solution

  Let's have a look at the given information:

  The snow gets one foot deeper every half an hour.

  This can be written as $\frac1{0.5}$. The snow gets $f$ feet deeper in two hours, which results in $\frac f2$.

  We can write the proportion:

  $\frac1{0.5}=\frac f2$

  Now we can cross multiply:

  $1\times 2=0.5\times f$

  In a last step, we can divide by $0.5$ to get the solution:

  $f=\frac2{0.5}=4$

• Solve the following proportions.

  Hints

  Cross-multiplication:

  $\frac ab=\frac cd \Leftrightarrow a\times d=c\times b$

  Depending on the value you want to determine, you will have to use the opposite operation.

  To solve the equation $2\times x=8$, you will have to divide by $2$ to get the solution $x=4$.

  You can check your result by plugging it into the given proportion.

  Solution

  You can use cross-multiplication to solve for unknown values in a proportion: $\frac ab=\frac cd$.

  Follow the steps:

  1. cross-multiply: $a\times d=c\times b$
  2. solve the equation by using opposite operations

  $\mathbf{\frac12=\frac4a}$

  • cross-multiply: $1\times a=4\times 2$
  • result is $a=8$
  • check: $\frac12=\frac48=\frac{4\div4}{8\div4}=\frac12$ $\surd$

  $\mathbf{\frac12=\frac b4}$

  • cross-multiply $1\times 4=b\times 2$
  • divide by $2$ to get the solution $b=4\div 2=2$
  • check: $\frac12=\frac24=\frac{2\div2}{4\div2}=\frac12$ $\surd$

  $\mathbf{\frac c2=\frac94}$

  • cross-multiply $c\times 4=9\times 2$
  • divide by $4$ and get $c=18\div 4=4.5$
  • check: $\frac{4.5}2=\frac{4.5\times 2}{2\times 2}=\frac94$ $\surd$

• Determine the number of days Freddy the yeti has to save his pocket money.

  Hints

  To figure this out, you can set up a proportion.

  You already know the proportion $\frac{3.5}7$

  • $3.5$ is his allowance for each week
  • $7$ is the number of days per week

  The number of days are he needs to save money are unknown.

  Represent the unknown number of days with a variable, and write it in the denominator.

  Cross-multiply to solve for the unknown value.

  $\frac ab=\frac cd \Leftrightarrow a\times d=c\times b$

  You can check the solution by plugging it into the proportion:

  $\frac {3.5}7=\frac ab$

  • $a$ is the amount of money needed to fulfill a wish
  • $b$ is the unknown number of days

  Solution

  Each proportion is written in this form:

  $\frac ab=\frac cd$

  You can check each solution by plugging it into the proportion:

  $\frac {3.5}7=\frac ab$

  • $a$ is the amount of money needed to fulfill a wish
  • $b$ is the unknown number of days

  Sledge

  • $\frac {3.5}7=\frac {14}b$
  • cross-multiplication results in $3.5 \times b=14\times 7$
  • dividing by $3.5$ gives us $b=28$
  • Freddy has to save his pocket money for $28$ days.

  CD

  • $\frac {3.5}7=\frac {5}b$
  • cross-multiplying results in $3.5 \times b=5\times 7$
  • dividing by $3.5$ gives us $b=10$
  • Freddy has to save his pocket money for $10$ days.

  Snow boots

  • $\frac {3.5}7=\frac {40}b$
  • cross-multiplying results in $3.5 \times b=40\times 7$
  • dividing by $3.5$ gives us $b=80$
  • Freddy has to save his pocket money for $80$ days.

  Ice cream

  • $\frac {3.5}7=\frac {3}b$
  • Cross-multiplication results in $3.5 \times b=3\times 7$
  • dividing by $3.5$ gives us $b=6$
  • Freddy has to save his pocket money for $6$ days.

• Find the correct proportion.

  Hints

  A proportion can be written as a fraction.

  For example, six scoops of ice-cream for three yetis can be written as $\frac63$.

  Keep in mind that for this ratio the amount of time is listed in the denominator.

  Solution

  What do we know?

  • The snow gets one foot deeper every half an hour. We can write this as a ratio: $\frac1{0.5}$.
  • In order to calculate the number of hours $h$ for the given depth of 3.5 feet we can write the ratio: $\frac {3.5}h$.

  We can now set up a proportion:

  $\frac1{0.5}=\frac {3.5}h$

  By cross-multiplying we can solve for the unknown value:

  $1\times h=0.5\times 3.5$

  The result is $h=1.75$. Andretti has to start in $1.75$ hours.

• Determine the number of hours till the snow is four feet deep.

  Hints

  Keep in mind that we have to solve two proportions.

  In the first proportion, the depth is unknown.

  In the second proportion, the number of hours is unknown.

  You have to subtract the depth of snow after two hours from four. This is the depth you have to use for the second equation.

  Keep in mind that you have to add two hours to the solution of the second equation.

  Solution

  To solve this problem, we will have to follow two steps:

  1. First of all we will have to determine the depth of the snow after two hours.
  2. If the snow isn't four feet deep yet, we will have to use a second proportion.

  Let's start:

  If the snow gets $0.25$ feet deeper every half an hour, how deep will the snow be after two hours?

  • first proportion: $\frac{0.25}{0.5}=\frac f2$
  • cross-multiplying results in $0.25\times 2=f\times 0.5$
  • divide by $0.5$ to get $f=1$

  After two hours, the snow is one foot deep.

  The snow isn't deep enough, so we'll have to determine how long it will take for three more feet of snow to fall ($4-1$).

  • second proportion: $\frac{0.75}{0.5}=\frac 3h$
  • cross-multiplication: $0.75\times h=3\times 0.5=1.5$
  • divide by $0.75$ to get $h=1.5\div 0.75=2$

  It will take another two hours to reach the depth of four ($3+1$) feet.

  Therefore we know that after four hours ($2+2$) the snow is four feet deep.