Q.If $0 < x < 1$, which of the following is greatest?

Comparing Fractions – The Intuition First

Imagine two pizzas of the same size. One is cut into 4 equal slices, and you take 3 of them — that's $\frac{3}{4}$ of a pizza. The other is cut into 8 equal slices, and you take 5 of them — that's $\frac{5}{8}$ of a pizza. Which one gives you more pizza?

Your eyes might tell you that 5 slices looks like more than 3 slices. But the slices themselves are different sizes. So you can't just compare the top numbers (numerators) — you have to account for how big each piece is. That's the whole problem of comparing fractions: the denominator tells you the size of each piece, and the numerator tells you how many of those pieces you have.

So comparing $\frac{3}{4}$ and $\frac{5}{8}$ is like comparing "3 big pieces" vs "5 smaller pieces." Which is more? You need a common language.

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The Precise Method: Same Denominator

The cleanest way to compare any two fractions is to rewrite them so they have the same denominator. Once the denominators match, you just compare the numerators — the fraction with the larger numerator is larger.

Example: Compare $\frac{3}{4}$ and $\frac{5}{8}$.

Find a common denominator. The smallest one that both 4 and 8 divide into is 8.

• $\frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8}$
• $\frac{5}{8}$ stays $\frac{5}{8}$

Now compare: $\frac{6}{8}$ vs $\frac{5}{8}$. Since $6 > 5$, we have $\frac{3}{4} > \frac{5}{8}$.

That matches the pizza intuition: three quarters of a pizza is indeed more than five eighths.

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The Cross-Multiplication Shortcut

You don't actually have to write the new denominators. Just compare the cross-products:

For $\frac{a}{b}$ and $\frac{c}{d}$:

• Compute $a \times d$ and $c \times b$.
• If $a \times d > c \times b$, then $\frac{a}{b} > \frac{c}{d}$.

Example: Compare $\frac{2}{3}$ and $\frac{3}{5}$.

Cross-multiply: $2 \times 5 = 10$, $3 \times 3 = 9$. Since $10 > 9$, we have $\frac{2}{3} > \frac{3}{5}$.

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Special Cases That Save Time

Same denominator: Just compare numerators. $\frac{7}{10} > \frac{3}{10}$ because $7 > 3$. …