Subtraction of Fractions: Worded problems

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 must calculate how much of the mountain Harry has not yet climbed after three days.

Let's break it down step-by-step:

• Step 1: Calculate the total amount of mountain climbed in three days.
  Harry climbs $\frac{1}{5}$ of the mountain each day. Therefore, in three days, he climbs:
  $\frac{1}{5} \times 3 = \frac{3}{5}$
• Step 2: Determine how much more Harry needs to climb.
  The whole mountain is represented by 1 (or $\frac{5}{5}$). To find out how much further Harry must climb, subtract the amount he has already climbed from the whole:
  $1 - \frac{3}{5} = \frac{5}{5} - \frac{3}{5} = \frac{2}{5}$

Therefore, the solution to the problem is that Harry must climb an additional $\frac{2}{5}$ of the mountain to reach the peak.

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

• Step 1: Identify the total fraction consumed by Roberto and Marcelo.
• Step 2: Find a common denominator to add these fractions.
• Step 3: Subtract the total consumed from the whole pizza.

Let's work through each step:

Step 1: Roberto eats $\frac{2}{7}$ of the pizza, and Marcelo eats $\frac{1}{3}$.

Step 2: To add these fractions, we need a common denominator. The least common denominator between 7 and 3 is 21.

Convert the fractions:
Roberto's fraction: $\frac{2}{7} = \frac{2 \times 3}{7 \times 3} = \frac{6}{21}$
Marcelo's fraction: $\frac{1}{3} = \frac{1 \times 7}{3 \times 7} = \frac{7}{21}$

Step 3: Add the converted fractions:
$\frac{6}{21} + \frac{7}{21} = \frac{13}{21}$

This result represents the total fraction of the pizza eaten by Roberto and Marcelo.

Step 4: Subtract the eaten fraction from the whole pizza (1, or $\frac{21}{21}$):
$\frac{21}{21} - \frac{13}{21} = \frac{8}{21}$

Therefore, the amount of pizza left for Diego to eat is $\frac{8}{21}$.

To solve this problem, we'll subtract $\frac{1}{12}$ from the full bottle (represented as '1' or $\frac{12}{12}$) for each hour for a total of 4 hours:

• Initially, the bottle is full: $\frac{12}{12}$.
• After 1 hour, subtract $\frac{1}{12}$:
  $\frac{12}{12} - \frac{1}{12} = \frac{11}{12}$.
• After 2 hours, subtract another $\frac{1}{12}$:
  $\frac{11}{12} - \frac{1}{12} = \frac{10}{12} = \frac{5}{6}$ after reduction.
• After 3 hours, subtract another $\frac{1}{12}$:
  $\frac{5}{6} - \frac{1}{12} = \frac{10}{12} - \frac{1}{12} = \frac{9}{12} = \frac{3}{4}$ after reduction.
• After 4 hours, subtract another $\frac{1}{12}$:
  $\frac{3}{4} - \frac{1}{12} = \frac{9}{12} - \frac{1}{12} = \frac{8}{12} = \frac{2}{3}$ after reduction.

Therefore, the amount of water that remains in the bottle after 4 hours is $\frac{2}{3}$.

Let's solve the problem step-by-step:

Step 1: Determine the total amount of cake eaten by adding $\frac{1}{4}$ and $\frac{1}{3}$.

Step 2: Find a common denominator for the fractions. The denominators 4 and 3 have a least common multiple of 12.

Step 3: Convert the fractions to have a denominator of 12:
$\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}$
$\frac{1}{3} = \frac{1 \times 4}{3 \times 4} = \frac{4}{12}$

Step 4: Add the fractions:
$\frac{3}{12} + \frac{4}{12} = \frac{7}{12}$

Step 5: Subtract the total eaten portion from the whole cake (1):
$1 - \frac{7}{12} = \frac{12}{12} - \frac{7}{12} = \frac{5}{12}$

Therefore, the fraction of the cake that is left is $\frac{5}{12}$.

To solve this problem, we will follow these steps:

• Step 1: Calculate the total fraction of the book that Janet reads over the two days.
• Step 2: Subtract this total from the whole to find the remaining fraction of the book.

Now, let's work through each step:

Step 1: Janet reads $\frac{1}{4}$ of the book on the first day and $\frac{1}{2}$ on the second day. First, we convert these fractions to have the same denominator before adding: $\frac{1}{4} + \frac{1}{2} = \frac{1}{4} + \frac{2}{4} = \frac{3}{4}$ This means Janet reads $\frac{3}{4}$ of the book in total.

Step 2: To determine how much of the book remains, we subtract this total from the whole book, which is represented by 1: $1 - \frac{3}{4} = \frac{4}{4} - \frac{3}{4} = \frac{1}{4}$ This calculation shows that Janet has $\frac{1}{4}$ of the book left to read.

Therefore, the solution to the problem is $\frac{1}{4}$.

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 $\frac{13}{12}$.

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.

To solve this problem, we need to determine how much of the cake is left after Jorge consumes part of it. We know the following:

• The cake initially has 8 slices, with only 3 slices left, meaning $\frac{3}{8}$ of the cake remains.
• Jorge eats $\frac{1}{4}$ of the entire original cake.

We need to find $\frac{1}{4}$ in terms of eighths to subtract it from $\frac{3}{8}$:

$\frac{1}{4} = \frac{2}{8}$ since multiplying both the numerator and the denominator by 2 converts it.

Thus, the amount of cake Jorge eats from the original amount available is equivalent to $\frac{2}{8}$ of the cake.

Now, we perform the subtraction to find the remaining portion of the cake:

$\frac{3}{8} - \frac{2}{8} = \frac{1}{8}$

Therefore, the amount of cake left after Jorge eats his portion is $\frac{1}{8}$ of the original cake. Thus, the final solution is:

$\frac{1}{8}$.