Question

Every January, Framingham Middle School holds its annual Winter Event. An article in the school paper reported that \(45 \%\) of seventh graders voted that this year's event should be an ice-skating party. The president of the seventh grade class said that \(\frac{9}{20}\) of seventh graders voted for ice skating. Could both reports be correct? Explain.

Short answer

Yes, both reports are correct because 45% is equivalent to \(\frac{9}{20}\).

Step-by-step solution

- Understand the Given Information

One source reports that 45% of seventh graders voted for the ice-skating party, while the president claims that \(\frac{9}{20}\) of them voted for it. We need to check if these two figures represent the same amount.

- Convert Percentage to Fraction

Convert 45% into a fraction. \ \ This is done by realizing that 45% means 45 out of 100, which is written as \(\frac{45}{100}\).

- Simplify the Fraction

Simplify \(\frac{45}{100}\) by dividing the numerator and the denominator by their greatest common divisor. The GCD of 45 and 100 is 5. Thus: \(\frac{45 \div 5}{100 \div 5} = \frac{9}{20}\).

- Compare the Fractions

Now compare the obtained fraction \(\frac{9}{20}\) with the president’s reported fraction, \(\frac{9}{20}\). Both fractions are equivalent.

Conclusion

Since \(\frac{9}{20}\) is equivalent to 45%, both reports can indeed be correct.

Key concepts

percentage to fraction conversion

Converting a percentage to a fraction is a common task in mathematics. It helps in comparing values more easily and understanding them in different forms.
For example, to convert 45% to a fraction, remember that 'percent' means 'per hundred'. So, 45% is the same as 45 out of 100. We write this as the fraction \(\frac{45}{100}\). This step makes it easier to work with and compare.

simplifying fractions

Once you have a fraction, simplifying it makes it easier to understand and use. Simplifying means reducing the fraction to its smallest form.
This is done by finding the greatest common divisor (GCD) of the numerator and the denominator. For example, the GCD of 45 and 100 is 5.
To simplify \(\frac{45}{100}\), divide both the numerator and the denominator by their GCD (5): \(\frac{45 \div 5}{100 \div 5} = \frac{9}{20}\). Now, \(\frac{9}{20}\) is in its simplest form.

fraction comparison

Comparing fractions can show us if two values are equivalent. After converting 45% to the fraction \(\frac{9}{20}\) and simplifying, we can compare it to the given fraction \(\frac{9}{20}\). In this case, both fractions are exactly the same, meaning the reported 45% is equivalent to the president’s \(\frac{9}{20}\).
Comparing fractions is helpful in verifying results and ensuring accuracy in reports.

greatest common divisor

Understanding the greatest common divisor (GCD) is crucial for simplifying fractions. The GCD of two numbers is the largest number that divides both without leaving a remainder.
In our example, the GCD of 45 and 100 is 5. To find this, you can list the factors of each number and choose the largest one they have in common.
For 45: 1, 3, 5, 9, 15, 45
For 100: 1, 2, 4, 5, 10, 20, 25, 50, 100
The common factors are 1 and 5. The largest is 5, so the GCD is 5.
This step is key in reducing fractions to their simplest form, aiding in easier calculation and comparison.