Fractions on a Number Line: Worded problems

To solve this problem, we'll compare each student's share of work against $\frac{2}{5}$ to determine who receives extra points.

Let's perform these comparisons:

• Daniel: He completed $\frac{1}{2}$ of the work.
  Compare $\frac{1}{2}$ with $\frac{2}{5}$:
  Cross-multiply: $1 \times 5$ vs $2 \times 2$ -> $5 > 4$.
  This shows $\frac{1}{2} > \frac{2}{5}$.
• Andy: He completed $\frac{1}{6}$ of the work.
  Compare $\frac{1}{6}$ with $\frac{2}{5}$:
  Cross-multiply: $1 \times 5$ vs $6 \times 2$ -> $5 < 12$.
  This shows $\frac{1}{6} < \frac{2}{5}$.
• Sara: She completed $\frac{2}{7}$ of the work.
  Compare $\frac{2}{7}$ with $\frac{2}{5}$:
  Cross-multiply: $2 \times 5$ vs $7 \times 2$ -> $10 < 14$.
  This shows $\frac{2}{7} < \frac{2}{5}$.

Only Daniel completed more than $\frac{2}{5}$ of the work. Thus, he receives extra points.

The teacher gives extra points to Daniel.

To solve this problem, we will compare the fractions $\frac{2}{7}$ and $\frac{3}{8}$ by finding a common denominator and then converting them to equivalent fractions:

• Step 1: Find the least common denominator (LCD) of 7 and 8. Since 7 and 8 are coprime (they have no common factors other than 1), the LCD is simply their product, $56$.
• Step 2: Convert each fraction to an equivalent fraction with the denominator of 56.
• To convert $\frac{2}{7}$ to a fraction with a denominator of 56: Multiply both the numerator and the denominator by 8 (since $7 \times 8 = 56$):
  $\frac{2}{7} = \frac{2 \times 8}{7 \times 8} = \frac{16}{56}$.
• To convert $\frac{3}{8}$ to a fraction with a denominator of 56: Multiply both the numerator and the denominator by 7 (since $8 \times 7 = 56$):
  $\frac{3}{8} = \frac{3 \times 7}{8 \times 7} = \frac{21}{56}$.
• Step 3: Compare the two fractions $\frac{16}{56}$ and $\frac{21}{56}$. Since 16 is less than 21, $\frac{2}{7}$ is less than $\frac{3}{8}$.

Therefore, Marcus eats less of the pizza than Silvia.

The correct answer is Marcus.

To solve this problem, we need to compare the fractions $\frac{1}{2}$ and $\frac{1}{4}$. Since both fractions have the same numerator, the fraction with the smaller denominator is the larger fraction, indicating which portion is larger.

Step-by-step, here's how it goes:

• Step 1: Consider the fractions $\frac{1}{2}$ (Andy's portion) and $\frac{1}{4}$ (Daniel's portion).
• Step 2: Compare these two fractions. Note that $\frac{1}{2}$ has a smaller denominator than $\frac{1}{4}$. In terms of fractions with the same numerator (1), a smaller denominator means a larger fraction.
• Step 3: Therefore, $\frac{1}{2}$ is greater than $\frac{1}{4}$, meaning Andy eats more than Daniel.

Consequently, the solution to the problem is that Andy eats more.

To solve this problem, we'll compare the fractions representative of the cake portions that Benjamin and George ate.

• Step 1: Simplify Fractions
  George ate $\frac{6}{9}$ of the cake. Simplifying this fraction, we divide both the numerator and the denominator by their greatest common divisor, which is 3. Thus, $\frac{6}{9} = \frac{2}{3}$.
• Step 2: Find Common Denominator
  The denominators of the two fractions we are comparing are 7 and 3. The least common multiple (LCM) of 7 and 3 is 21.
• Step 3: Convert to Common Denominator
  Convert $\frac{4}{7}$ and $\frac{2}{3}$ to fractions with a common denominator of 21:
  - For $\frac{4}{7}$, multiply the numerator and the denominator by 3: $\frac{4}{7} = \frac{4 \times 3}{7 \times 3} = \frac{12}{21}$.
  - For $\frac{2}{3}$, multiply the numerator and the denominator by 7: $\frac{2}{3} = \frac{2 \times 7}{3 \times 7} = \frac{14}{21}$.
• Step 4: Compare the Fractions
  Now, compare the numerators: 12 and 14. Since 14 is greater than 12, we can conclude that $\frac{14}{21}$ (or $\frac{2}{3}$) is greater than $\frac{12}{21}$ (or $\frac{4}{7}$).

Therefore, George ate more of the cake.

The correct answer is George.