Understanding the Pizza Fraction Problem

This problem asks us to find the fraction of a pizza that is left after three different people eat portions of it in sequence. Each person eats a fraction of what was remaining before they took their share.

Step-by-Step Calculation of Pizza Fractions Left

Let's represent the whole pizza as the number 1.

1. Pizza Fraction After Raju Eats

Raju ate \(\frac{3}{8}\) part of the whole pizza.

Fraction of pizza remaining after Raju = Whole pizza - Raju's part

Remaining after Raju = \(1 - \frac{3}{8}\)

To subtract, we find a common denominator, which is 8. \(1 = \frac{8}{8}\).

Remaining after Raju = \(\frac{8}{8} - \frac{3}{8} = \frac{8 - 3}{8} = \frac{5}{8}\)

So, \(\frac{5}{8}\) of the pizza was left after Raju ate his part.

2. Pizza Fraction After Adam Eats

Adam ate \(\frac{3}{10}\) part of the remaining pizza. The remaining pizza at this point was \(\frac{5}{8}\).

Fraction Adam ate = \(\frac{3}{10}\) of \(\frac{5}{8}\)

Fraction Adam ate = \(\frac{3}{10} \times \frac{5}{8} = \frac{3 \times 5}{10 \times 8} = \frac{15}{80}\)

We can simplify this fraction by dividing the numerator and denominator by their greatest common divisor, which is 5.

Fraction Adam ate = \(\frac{15 \div 5}{80 \div 5} = \frac{3}{16}\)

Adam ate \(\frac{3}{16}\) of the whole pizza.

Now, let's find the fraction of pizza remaining after Adam ate. This is the amount remaining after Raju minus the amount Adam ate.

Remaining after Adam = Remaining after Raju - Adam's part

Remaining after Adam = \(\frac{5}{8} - \frac{3}{16}\)

To subtract, we find a common denominator, which is 16. \(\frac{5}{8} = \frac{5 \times 2}{8 \times 2} = \frac{10}{16}\).

Remaining after Adam = \(\frac{10}{16} - \frac{3}{16} = \frac{10 - 3}{16} = \frac{7}{16}\)

So, \(\frac{7}{16}\) of the pizza was left after Adam ate his part.

3. Pizza Fraction After Renu Eats

Renu ate \(\frac{4}{7}\) part of the pizza that was left. The pizza left at this point was \(\frac{7}{16}\).

Fraction Renu ate = \(\frac{4}{7}\) of \(\frac{7}{16}\)

Fraction Renu ate = \(\frac{4}{7} \times \frac{7}{16} = \frac{4 \times 7}{7 \times 16} = \frac{28}{112}\)

We can simplify this fraction by dividing the numerator and denominator by their greatest common divisor, which is 28.

Fraction Renu ate = \(\frac{28 \div 28}{112 \div 28} = \frac{1}{4}\)

Renu ate \(\frac{1}{4}\) of the whole pizza.

Finally, let's find the fraction of pizza that is still left after Renu ate. This is the amount remaining after Adam minus the amount Renu ate.

Fraction still left = Remaining after Adam - Renu's part

Fraction still left = \(\frac{7}{16} - \frac{1}{4}\)

To subtract, we find a common denominator, which is 16. \(\frac{1}{4} = \frac{1 \times 4}{4 \times 4} = \frac{4}{16}\).

Fraction still left = \(\frac{7}{16} - \frac{4}{16} = \frac{7 - 4}{16} = \frac{3}{16}\)

Summary of Pizza Fractions

The fraction of the pizza that is still left is \(\frac{3}{16}\).

Revision Table: Key Concepts in Fraction Problems

Additional Information on Sequential Fraction Calculations

When solving problems where fractions are taken from a remaining amount, it's crucial to correctly identify the base quantity for each step. The second person's fraction is not of the original whole unless stated, but of the amount left after the first person.

Let's look at the total fraction of pizza eaten by everyone:

• Raju ate: \(\frac{3}{8}\)
• Adam ate: \(\frac{3}{16}\)
• Renu ate: \(\frac{1}{4}\)

Total fraction eaten = Raju's part + Adam's part + Renu's part

Total eaten = \(\frac{3}{8} + \frac{3}{16} + \frac{1}{4}\)

Find a common denominator, which is 16.

• \(\frac{3}{8} = \frac{3 \times 2}{8 \times 2} = \frac{6}{16}\)
• \(\frac{1}{4} = \frac{1 \times 4}{4 \times 4} = \frac{4}{16}\)

Total eaten = \(\frac{6}{16} + \frac{3}{16} + \frac{4}{16} = \frac{6 + 3 + 4}{16} = \frac{13}{16}\)

The fraction left is the whole pizza minus the total eaten:

Fraction left = \(1 - \frac{13}{16} = \frac{16}{16} - \frac{13}{16} = \frac{16 - 13}{16} = \frac{3}{16}\)

This confirms our step-by-step calculation is correct. The two methods (calculating remaining at each step vs. calculating total eaten) yield the same result for the fraction of pizza left.