Comparing Simple Fractions

Which is greater: $\frac{3}{4}$ or $\frac{5}{7}$?

Set up the cross multiplication: $\frac{3}{4}$ vs $\frac{5}{7}$ = Ready to cross multiply

Multiply first numerator by second denominator: $3 \times 7 = 21$ = Left product: $21$

Multiply second numerator by first denominator: $5 \times 4 = 20$ = Right product: $20$

Compare the products: $21 > 20$, so $\frac{3}{4} > \frac{5}{7}$ = $\frac{3}{4}$ is greater

Answer: $\frac{3}{4} > \frac{5}{7}$

Fractions That Are Equal

Compare $\frac{4}{6}$ and $\frac{6}{9}$.

Set up the cross multiplication: $\frac{4}{6}$ vs $\frac{6}{9}$ = Ready to cross multiply

Calculate left product: $4 \times 9 = 36$ = Left product: $36$

Calculate right product: $6 \times 6 = 36$ = Right product: $36$

Compare the products: $36 = 36$, so $\frac{4}{6} = \frac{6}{9}$ = The fractions are equal

Answer: $\frac{4}{6} = \frac{6}{9}$ (both simplify to $\frac{2}{3}$)

Comparing with Larger Numbers

Which is smaller: $\frac{7}{12}$ or $\frac{5}{9}$?

Set up the cross multiplication: $\frac{7}{12}$ vs $\frac{5}{9}$ = Ready to cross multiply

Calculate left product: $7 \times 9 = 63$ = Left product: $63$

Calculate right product: $5 \times 12 = 60$ = Right product: $60$

Compare the products: $63 > 60$, so $\frac{7}{12} > \frac{5}{9}$ = $\frac{5}{9}$ is smaller

Answer: $\frac{5}{9} < \frac{7}{12}$

Real-World Application

Two pizzas were shared. In Pizza A, 3 out of 8 slices remain. In Pizza B, 2 out of 5 slices remain. Which pizza has more left proportionally?

Write as fractions: Pizza A: $\frac{3}{8}$, Pizza B: $\frac{2}{5}$ = Fractions identified

Cross multiply: $3 \times 5 = 15$ and $2 \times 8 = 16$ = Products: $15$ vs $16$

Compare: $15 < 16$, so $\frac{3}{8} < \frac{2}{5}$ = $\frac{2}{5}$ is greater

Answer the question: Pizza B has $\frac{2}{5}$ remaining, which is greater = Pizza B has more left

Answer: Pizza B has more pizza left proportionally.

Mistake: Multiplying numerator by numerator instead of cross multiplying

Why: Students sometimes confuse cross multiplication with regular multiplication. Cross multiplication means diagonally: numerator of first with denominator of second.

Correct: Always multiply across: numerator of one fraction times denominator of the other.

Mistake: Mixing up which product corresponds to which fraction

Why: After calculating both products, students forget which product belongs to which fraction.

Correct: The product using the first fraction's numerator tells you about the first fraction. Write the products directly below or beside each fraction.

Mistake: Thinking cross multiplication gives the actual fraction value

Why: Students may think the products are the fractions' values. They are not; they are only useful for comparison.

Correct: Cross multiplication tells you which fraction is larger, not what the fractions equal as decimals or in lowest terms.

Shopping Comparisons

Stores often display different package sizes at different prices. Cross multiplication helps find the better deal.

If a 3-pack costs 8 euros and a 5-pack costs 12 euros, compare $\frac{3}{8}$ and $\frac{5}{12}$ (items per euro). Cross multiply: $3 \times 12 = 36$ vs $5 \times 8 = 40$. Since $36 < 40$, you get more items per euro with the 5-pack.

Recipe Scaling

When comparing recipes, cross multiplication helps determine which uses more of an ingredient proportionally.

Recipe A uses $\frac{2}{3}$ cup of sugar for 12 cookies. Recipe B uses $\frac{3}{5}$ cup for 12 cookies. Which is sweeter? Cross multiply: $2 \times 5 = 10$ vs $3 \times 3 = 9$. Since $10 > 9$, Recipe A uses more sugar.

Q: Why does cross multiplication work?

A: Cross multiplication is actually finding a common denominator behind the scenes. When you compare $\frac{a}{b}$ and $\frac{c}{d}$, you could rewrite them as $\frac{a \times d}{b \times d}$ and $\frac{c \times b}{d \times b}$. Since both denominators become $b \times d$, you only need to compare the numerators: $a \times d$ vs $c \times b$.

Q: Can I use cross multiplication with more than two fractions?

A: Cross multiplication compares exactly two fractions at a time. For three or more fractions, compare them in pairs, or convert all fractions to the same denominator or to decimals.

Q: Does the order matter?

A: The order determines which product belongs to which fraction. Be consistent: the product using the first fraction's numerator corresponds to the first fraction. If you switch the fractions, switch your interpretation too.