Find the Total Students: Solving the 1/5 and 20% Participation Problem

Fractional Equations with Mixed Percentages

During recess 15 \frac{1}{5} of the students play catch, 20% play soccer and the remaining 15 students watch a movie.

How many students are there in total?

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Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

During recess 15 \frac{1}{5} of the students play catch, 20% play soccer and the remaining 15 students watch a movie.

How many students are there in total?

2

Step-by-step solution

To solve this problem, we will find a common equation to account for all students:

  • Activity: Playing catch, Fraction: 15x\frac{1}{5}x
  • Activity: Playing soccer, Fraction: 20% of xx which is 15x\frac{1}{5}x
  • Activity: Watching a movie, Number: 1515 students

The total number of students involved is xx. Thus, the setup for the equation is:

15x+15x+15=x\frac{1}{5}x + \frac{1}{5}x + 15 = x

Simplify and solve for xx:

25x+15=x\frac{2}{5}x + 15 = x

Subtract 25x\frac{2}{5}x from both sides:

15=x25x15 = x - \frac{2}{5}x

15=35x15 = \frac{3}{5}x

To isolate xx, multiply both sides by 53\frac{5}{3}:

x=15×53x = 15 \times \frac{5}{3}

x=25x = 25

Therefore, the total number of students is 25 students.

3

Final Answer

25 students

Key Points to Remember

Essential concepts to master this topic
  • Setup: Convert all parts to fractions before writing equation
  • Technique: Convert 20% to 15 \frac{1}{5} to match 15 \frac{1}{5}
  • Check: Verify 15(25)+15(25)+15=25 \frac{1}{5}(25) + \frac{1}{5}(25) + 15 = 25

Common Mistakes

Avoid these frequent errors
  • Treating 20% as 20 instead of 0.20
    Don't write 20% as 20 in your equation = massive overcount! This makes the equation impossible to solve correctly. Always convert percentages to fractions or decimals first: 20% = 15 \frac{1}{5} = 0.20.

Practice Quiz

Test your knowledge with interactive questions

Convert the fraction into a percentage:

\( \frac{2}{100}=\text{?} \)

FAQ

Everything you need to know about this question

Why does 20% equal 1/5?

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20% means 20 out of 100, which equals 20100=15 \frac{20}{100} = \frac{1}{5} . This makes the problem easier since both fractions have the same denominator!

What if I use decimals instead of fractions?

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That works too! Use 0.2x + 0.2x + 15 = x. Just be careful with decimal arithmetic and consider using fractions to avoid rounding errors.

How do I know which students are the 'remaining' ones?

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The remaining students are those not counted in the first two activities. Since 15+15=25 \frac{1}{5} + \frac{1}{5} = \frac{2}{5} are accounted for, the remaining 35 \frac{3}{5} must be the 15 movie watchers.

Can I solve this without setting up an equation?

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You could try guess-and-check, but setting up an equation is much more reliable. It shows your thinking clearly and works for any similar problem, no matter the numbers.

What if my fractions don't add up to the total?

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Double-check your setup! Make sure you've correctly identified what each fraction represents and that all students are accounted for: playing + watching + other activities = total.

Why multiply both sides by 5/3 at the end?

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To isolate x from 35x=15 \frac{3}{5}x = 15 , we need to divide by the coefficient 35 \frac{3}{5} . Dividing by a fraction is the same as multiplying by its reciprocal: 53 \frac{5}{3} !

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