Factors and Fractions

Comparing Rational Numbers

Compare two fractions using cross-multiplication. Multiply each numerator by the other denominator and compare the products.

Example

Compare two fractions by cross multiplication. Cross multiplication preserves the inequality only when both denominators are positive; write any negative sign in the numerator first.

highlighted = computed this step

Step 1 — Set up

Set up the expression.

2334\frac{2}{3} \quad \frac{3}{4}
Start with 2/3 and 3/4; the cuts are different.2/33/4Use a shared denominator: 3 x 4 = 12.

Step 2 — Left cross product

Cross multiply left: 2 x 4 = 8.

2×4=82 \times 4 = \hl{8}
Convert the left fraction to denominator 12: 2 x 4 = 8.8/123/4 next2/3 = 8/12; next align 3/4Same denominator means the larger numerator is farther to the right.

Step 3 — Right cross product

Cross multiply right: 3 x 3 = 9.

3×3=93 \times 3 = \hl{9}
Convert the right fraction to denominator 12: 3 x 3 = 9.8/129/122/3 = 8/12; 3/4 = 9/12Same denominator means the larger numerator is farther to the right.

Step 4 — Compare products

Compare products: 8 < 9.

8<9\hl{8} < \hl{9}
Now compare aligned numerators: 8 < 9.8/129/122/3 = 8/12; 3/4 = 9/12Same denominator means the larger numerator is farther to the right.

Step 5 — Result

The left fraction is less than the right fraction.

23<34\frac{2}{3} < \frac{3}{4}
So 2/3 < 3/4.8/129/122/3 = 8/12; 3/4 = 9/12Same denominator means the larger numerator is farther to the right.
compare-rationals To compare fractions a/b and c/d: - Cross-multiply: LHS = a·d, RHS = c·b. - If LHS < RHS then a/b < c/d.