Solving Percent Equations

Solving Percent Equations exercise

• Describe how you can determine the percent of honey.

  Hints

  Determine which term is unknown. We know the quantity of love potion as well as the percent values.

  Usually we use the letter $x$ for the unknown variable.

  Solution

  Richard has fallen in love with Princess Head Over Heals, but she doesn't like him one bit. So, using his grandmother's recipe, he wants to concoct a love potion.

  The ingredients for $200~ml$ of love potion are:

  • $20~\%$ honey,
  • $\frac12$ batwing,
  • $50$ teardrops of love and
  • $17.5~\%$ horn of a unicorn.

  Let's help Richard figure out just how much honey he needs to make the love potion.

  We can use the equation $\frac{\text{part}}{\text{whole}}=\frac{\text{percent}}{100}$.

  The whole amount shall be $200~ml$, and he needs $20$ percent of honey. First of all we plug in the known values as well as the variable $x$ for the unknown part and get: $\frac x{200}=\frac{20}{100}$.

  We can use cross multiplication:

  $100\times x =20\times 200$.

  Simplifying leads to

  $100x=4000$

  and dividing by $100$ (opposite operation of multiplication) gives us the solution: Richard needs $40~ml$ of honey for his potion.

• Calculate how much is $17.5~\%$ of $200~ml$.

  Hints

  Decide what's the part, the whole, and percent.

  The percent is given in the story above.

  To make the recipe, we need unicorn horn extract and some other ingredients. Unicorn horn extract is a part of the potion, so it's quantity is a part of the whole quantity.

  To cross multiply equal fractions or ratios, multiply the numerator of each side by the denominator of the other side and set the two products equal.

  To isolate a variable use opposite operations:

  • $+\longleftrightarrow -$ and
  • $\times \longleftrightarrow \div$.

  Solution

  How many $ml$ are in $17.5~\%$ of $200~ml$. This is the quantity of unicorn horn extract Richard needs to make the love potion.

  Use the equation $\frac{\text{part}}{\text{whole}}=\frac{\text{percent}}{100}$.

  The whole amount is $200~ml$, and the percent is $17.5$. Now we can substitute the part by $x$, the whole by $200$, and the percent by $17.5$ to get the following equation:

  $\frac x{200}=\frac{17.5}{100}$.

  Cross multiply to get: $100\times x =17.5\times 200$.

  Simplifying give us: $100x=3500$.

  Now we use the opposite operation of multiplication: division. We divide both sides by $100$ and get the result $x=35$. Richard needs $35~ml$ of unicorn horn extract to make the love potion.

• Define the whole, the part, and the percentage.

  Hints

  The percentage can be $100$ at most.

  The whole is the total quantity.

  You can recognize percent by the $\%$ sign.

  The whole is more than each part.

  Solution

  Remember the equation $\frac{\text{part}}{\text{whole}}=\frac{\text{percent}}{100}$.

  It's very important to decide which part is the whole, which is the part and finally what the percentage is.

  Let's take a look at the new recipe for the love potion.

  • The total quantity $300$ is the whole.
  • This whole is made up of different ingredients. If these are also given in $ml$ we talk about parts. So $60~ml$ honey is the part.
  • We can recognize the percent by the $\%$ sign. So $40~\%$ unicorn horn extract is a percent.

  Now let's have a look at the potion for everlasting youth.

  • What is the whole? It's the total amount of $400$.
  • And what are the parts? The parts are given in the same unit as the whole: $100~ml$ of bat ear extract and $120~ml$ of unicorn tears.
  • You will finde the percentage by looking for the $\%$ sign: $20$ $\%$ of honey and $25$ $\%$ of milk.

• Help Richard convert the percentage into $ml$.

  Hints

  You can add all the results. The sum is $200$.

  The whole is the total quantity of the potion.

  Solution

  Oh, all the ingredients are given in percent.

  Richard is a little bit helpless. He needs to know exactly how many $ml$ he needs of each ingredient.

  Convert the equation:

  $\frac{\text{part}}{\text{whole}}=\frac{\text{percent}}{100}$

  by cross multiplying:

  $100\times x=\text{whole}\times \text{percent}$.

  Dividing by $100$ will give Richard the information he needs:

  $x=\text{whole}\times \text{percent}\div 100$.

  1. Dragon eye extract: $200\times 10\div100=20$ $ml$.
  2. Dragon tears: $200\times 30\div100=60$ $ml$.
  3. Eagle feather extract: $200\times 20\div100=40$ $ml$.
  4. Honey: $200\times 40\div100=80$ $ml$.

• Explain the meaning of percent.

  Hints

  $100$ cents equals one dollar.

  One cent equals one hundredth of one dollar.

  We can write percent as a decimal number: $20~\%=0.20$.

  We can write a decimal number as percent by multiplying by $100$ and writing the $\%$ sign behind: $0.175=17.5~\%$.

  Solution

  Richard the mage needs $20~\%$ of honey and $17.5~\%$ unicorn horn extract to make the love potion. But what does this mean?

  If the love potion is divided into $100$ equal parts then each part will represent one percent. Percent means one over hundred.

  In this way $20~\%$ is the same as $20$ parts of the whole. So we have to divide by $100$ and multiply with $20$: $\frac{20}{100}=0.20$. You can also write: $100\times 0.20=20$. This is the number of parts of the potion.

  In general we can use the equation $\frac{\text{part}}{\text{whole}}=\frac{\text{percent}}{100}$.

• Determine the percentage.

  Hints

  Use cross multiplication to solve for unknown quantities or check that the ratios are equal.

  Find the percentage by multiplying $\large{\frac{\text{part}}{\text{whole}}}$ by $100$.

  Solution

  Using the equation $\frac{\text{part}}{\text{whole}}=\frac{\text{percent}}{100}$.

  We can either substitute the given values and transform this equation or transform this equation and substitute afterwards. This time the unknown is the percentage, so we assign the variable $x$ to the unknown value.

  Cross multiplying leads to:

  $100\times \text{part}=\text{whole}\times x$.

  Now we divide by the whole and get:

  $x=100 \times \text{part}\div \text{whole}$.

  • $10$ parts of $100$ are $100\times 10\div 100=10~\%$.
  • $100~ml$ of $400~ml$ are $100 \times 100 \div 400 = 25 ~\%$.
  • $360$ miles in respect of $1200$ miles are $100 \times 360 \div 1200=30~\%$.
  • $270~ml$ of $600~ml$ are $100 \times 270 \div 600=45~\%$.