Let's break down the problem step-by-step to find out how many pizzas were left in the end.
1. Initial Number of Pizzas:
- Hanny initially made 7 pizzas.
2. Pizzas Given to the Neighbor:
- Hanny gave [tex]$2 \frac{1}{2}$[/tex] pizzas to her neighbor.
- This can be written as a mixed number: [tex]\(2 \frac{1}{2} = 2 + \frac{1}{2} = \frac{4}{2} + \frac{1}{2} = \frac{5}{2}\)[/tex].
3. Pizzas Eaten by Her Children:
- Each of her 3 children ate [tex]$\frac{1}{4}$[/tex] pizza.
- The total amount eaten by her 3 children is [tex]\(3 \times \frac{1}{4} = \frac{3}{4} \)[/tex].
4. Pizzas Left After Giving to the Neighbor and Children Eating:
- Calculate the total taken away by the neighbor and the children: [tex]\(\frac{5}{2} + \frac{3}{4}\)[/tex].
- Convert [tex]$\frac{5}{2}$[/tex] to a common denominator with [tex]$\frac{3}{4}$[/tex]: [tex]\( \frac{5}{2} = \frac{10}{4} \)[/tex].
- Thus, [tex]\(\frac{10}{4} + \frac{3}{4} = \frac{13}{4} \)[/tex].
- Subtract this from the initial number of pizzas: [tex]\(7 - \frac{13}{4}\)[/tex].
- Convert 7 to a fraction with the same denominator: [tex]\( 7 = \frac{28}{4} \)[/tex].
- Therefore, [tex]\(\frac{28}{4} - \frac{13}{4} = \frac{15}{4} \)[/tex].
5. Addition of More Pizzas:
- After this, Hanny made an additional 4 pizzas.
- Convert 4 to a fraction with the same denominator: [tex]\( 4 = \frac{16}{4} \)[/tex].
- Add these to the current total: [tex]\(\frac{15}{4} + \frac{16}{4} = \frac{31}{4} \)[/tex].
6. Fraction of Pizzas That Broke:
- Out of the total pizzas, [tex]$\frac{3}{5}$[/tex] of them broke.
- Calculate [tex]$\frac{3}{5}$[/tex] of [tex]\(\frac{31}{4}\)[/tex]: [tex]\( \frac{3}{5} \times \frac{31}{4} = \frac{93}{20} \)[/tex].
7. Pizzas Left After Discarding Broken Ones:
- Subtract the broken pizzas from the total number: [tex]\(\frac{31}{4} - \frac{93}{20}\)[/tex].
- Convert [tex]\(\frac{31}{4}\)[/tex] to a common denominator with [tex]\(\frac{93}{20}\)[/tex]: [tex]\( \frac{31}{4} = \frac{155}{20} \)[/tex].
- Therefore, [tex]\(\frac{155}{20} - \frac{93}{20} = \frac{62}{20} \)[/tex].
8. Simplify the Resulting Fraction:
- Simplify [tex]\(\frac{62}{20}\)[/tex] by dividing the numerator and the denominator by their greatest common divisor (GCD), which is 2: [tex]\( \frac{62 \div 2}{20 \div 2} = \frac{31}{10} \)[/tex].
9. Convert to a Mixed Number:
- Convert [tex]\(\frac{31}{10}\)[/tex] to a mixed number.
- Divide 31 by 10: [tex]\( 31 \div 10 = 3 \)[/tex] remainder [tex]\( 1 \)[/tex].
- So, [tex]\(\frac{31}{10} = 3 \frac{1}{10} \)[/tex].
Therefore, the final answer is that there are [tex]\(3 \frac{1}{10}\)[/tex] pizzas left in the end.
