Subtraction of Fractions: Worded problems

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'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 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 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 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 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 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:

$\frac{1}{8}$.