All Operations in Fractions: Worded problems

To solve this problem, we need to add the fractions of the packet that Dana eats over the three days.

Let's outline our steps:

• Step 1: Find the common denominator for the fractions $\frac{1}{2}$, $\frac{1}{3}$, and $\frac{1}{4}$.
• Step 2: Convert each fraction to an equivalent fraction with the common denominator.
• Step 3: Add the fractions together.

Step 1: Determine the least common denominator (LCD).
The denominators are 2, 3, and 4. The least common multiple (LCM) of these numbers is 12. Hence, the common denominator is 12.

Step 2: Convert each fraction:
- $\frac{1}{2} = \frac{1 \times 6}{2 \times 6} = \frac{6}{12}$
- $\frac{1}{3} = \frac{1 \times 4}{3 \times 4} = \frac{4}{12}$
- $\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}$

Step 3: Add the fractions:
$\frac{6}{12} + \frac{4}{12} + \frac{3}{12} = \frac{6+4+3}{12} = \frac{13}{12}$

The amount Dana eats over the three days is $\frac{13}{12}$ of the packet.

This means that Dana ate more than a whole packet (since $\frac{13}{12}$ is more than 1).

Therefore, the solution to the problem is $\frac{13}{12}$.

The mother spends time preparing both salad and french fries. We need to sum up these two times:

Given the times:

• Salad: $\frac{1}{6}$ hour
• French fries: $\frac{2}{3}$ hour

To add these fractions, we need a common denominator. The denominators are 6 and 3. The least common multiple of 6 and 3 is 6.

Convert $\frac{2}{3}$ to a fraction with a denominator of 6:

$\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}$

Now, add the fractions:

$\frac{1}{6} + \frac{4}{6} = \frac{1 + 4}{6} = \frac{5}{6}$

Therefore, the total time the mother spends preparing food is $\frac{5}{6}$ of an hour.

To solve this problem, we'll follow these steps:

• Step 1: Determine how much of the original amount remains after taking out $\frac{2}{7}$.
• Step 2: Calculate how much more needs to be removed so that only half of the original amount remains in the piggy bank.
• Step 3: Perform the necessary calculations to find the additional amount required to be taken out.

Now, let's work through each step:

Step 1: Initially, Marcos takes out $\frac{2}{7}$ of his money. Therefore, the remaining money is:

$1 - \frac{2}{7} = \frac{7}{7} - \frac{2}{7} = \frac{5}{7}$

Step 2: We want only half of the initial amount, or $\frac{1}{2}$, to remain. Let $x$ be the additional fraction of money taken out.

Equation: $\frac{5}{7} - x = \frac{1}{2}$

Step 3: Solve for $x$. First, get a common denominator for the fractions on the right.

$\frac{5}{7} - x = \frac{1}{2}$

Find a common denominator (here, 14 works):

$\frac{5 \times 2}{7 \times 2} - x = \frac{1 \times 7}{2 \times 7}$

$\frac{10}{14} - x = \frac{7}{14}$

Subtracting $\frac{7}{14}$ from both sides gives us:

$\frac{10}{14} - \frac{7}{14} = x$

Hence, $x = \frac{3}{14}$.

Therefore, the solution to the problem is that Marcos needs to take out an additional amount of $\frac{3}{14}$ of his money.

To solve this problem, we will add the fractions representing the amount of paper Daniel uses:

• Step 1: Identify the fractions of the paper roll used.
  Daniel uses $\frac{2}{5}$ of the roll for the book and $\frac{3}{4}$ for the notebook.
• Step 2: Find a common denominator.
  The denominators are 5 and 4. The least common multiple of 5 and 4 is 20. Hence, the common denominator will be 20.
• Step 3: Convert each fraction to have this common denominator.
  • For $\frac{2}{5}$, convert it to $\frac{2}{5} \times \frac{4}{4} = \frac{8}{20}$.
  • For $\frac{3}{4}$, convert it to $\frac{3}{4} \times \frac{5}{5} = \frac{15}{20}$.
• Step 4: Add the two fractions.
  Add $\frac{8}{20} + \frac{15}{20} = \frac{23}{20}$.
• Step 5: Interpret the result.
  The result $\frac{23}{20}$ implies that Daniel used more than one whole roll, specifically $\frac{23}{20}$ of the paper roll in total.

Therefore, the solution to the problem is $\frac{23}{20}$.

To solve the problem of determining how long Ned spends doing his homework, we need to add the times he spent on language and science homework.

Given:
- Language homework: $\frac{1}{8}$ of an hour
- Science homework: $\frac{6}{8}$ of an hour

Since both times have the same denominator, adding them is straightforward:

$\frac{1}{8} + \frac{6}{8} = \frac{1+6}{8} = \frac{7}{8}$

Therefore, Ned spends $\frac{7}{8}$ of an hour doing his homework.

To solve this problem, we will add the fractions of the work Sarah completed in the first and second hours:

• Step 1: Identify the fractions: Sarah completed $\frac{2}{8}$ of the work in the first hour and $\frac{1}{4}$ of the work in the second hour.
• Step 2: Convert the fractions to have the same denominator. The denominators are 8 and 4, respectively. The least common denominator (LCD) of 8 and 4 is 8.
• Step 3: Convert $\frac{1}{4}$ to a fraction with a denominator of 8. Since $\frac{1}{4} = \frac{2}{8}$, we convert it as follows: $\frac{1}{4} = \frac{1 \times 2}{4 \times 2} = \frac{2}{8}$
• Step 4: Add the fractions with the common denominator: $\frac{2}{8} + \frac{2}{8} = \frac{2 + 2}{8} = \frac{4}{8}$
• Step 5: Simplify the resulting fraction. Divide the numerator and the denominator by their greatest common divisor, which is 4: $\frac{4}{8} = \frac{4 \div 4}{8 \div 4} = \frac{1}{2}$

Therefore, Sarah completed $\frac{1}{2}$ of the assignment in total.