Solve the following

Explanation

We will look for the key words is, of, and what number in the percent sentence and translate them to mathematical symbols to form a percent equation.  Then we can solve the equation to find the unknown number.

First, we translate

31.5 is 33\(\frac13\) of what number?

31.5 = 33\(\frac13\)% • x

In this case, the calculations can be made easier by writing 33\(\frac13\)% as a fraction instead of as a repeating decimal.

• Recall that 6.1 that 33\(\frac13\)% = \(\frac13\)
  • 31.5 = \(\frac13 \cdot x\)
• To undo the multiplication by \(\frac13\) and isolate x on the right side of the equation, divide both sides by \(\frac13\).
  • \(\LARGE{\frac{31.5}{\color{red}{\frac13}} = \frac{\frac13 \cdot x}{\color{red}{\frac13}}}\)
• To simplify the fraction on the right side of the equation, remoe the common factor of \(\frac13\) from the numerator and denominator.
  • \(\require{cancel} \LARGE{\frac{31.5}{\frac13} = \frac{\cancel{\frac13}^1 \cdot x}{\cancel{\frac13}_1}}\)
• On the left side, the fraction bar indicates division.
  • \(31.5 \color{red}{÷ \frac13} = x\)
• On the left side, write 31.5 as a fraction: \(\frac{31.5}{1}\).  Then use the rule for dividing fractions:  Multiply by the reciprocal of \(\frac13\), which is \(\frac31\).  Do the multiplication.
  • \(\frac{31.5}{1} \cdot \frac31 = x\)
• Do the multiplication
  • 94.5 = x

Thus, 31.5 is 33\(33\frac13\)% of 94.5