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when do you cross multiply fractions

When Do You Cross Multiply Fractions? (Quick Scoop)

You cross multiply fractions mainly when you are dealing with **proportions** (a fraction equal to a fraction) or when you are **comparing two fractions** to see which is larger.

What “Cross Multiply” Actually Means

When you have two fractions, like ab\frac{a}{b}ba​ and cd\frac{c}{d}dc​, cross multiplying means:
  • Multiply the top of the first fraction by the bottom of the second: a×da\times da×d.
  • Multiply the top of the second fraction by the bottom of the first: c×bc\times bc×b.

You are literally drawing an invisible “X” between numerator and denominator across the equals or comparison sign.

1. When You SHOULD Cross Multiply

Case A: Fraction = Fraction (Proportions)

This is the classic situation:

ab=cd\frac{a}{b}=\frac{c}{d}ba​=dc​

Here, cross multiplication lets you remove the fractions and solve for a variable.

You do:

a×d=b×ca\times d=b\times ca×d=b×c

Then solve the resulting equation. Example (Solving a Proportion)
Solve 23=x9\frac{2}{3}=\frac{x}{9}32​=9x​.

  1. Cross multiply: 2×9=3×x2\times 9=3\times x2×9=3×x.
  1. So 18=3x18=3x18=3x.
  2. Then x=6x=6x=6.

Any time your problem looks like “one fraction equals another fraction” (a proportion), cross multiplication is a natural, efficient move.

Case B: Comparing Two Fractions

Suppose you want to know which is bigger: 37\frac{3}{7}73​ or 58\frac{5}{8}85​. The denominators are different (unlike fractions), so cross multiplying is a quick comparison trick.

Steps:

  1. Multiply the numerator of the first fraction by the denominator of the second:
    3×8=243\times 8=243×8=24.
  1. Multiply the numerator of the second by the denominator of the first:
    5×7=355\times 7=355×7=35.
  1. Compare 24 and 35. Since 24 < 35, that means 37<58\frac{3}{7}<\frac{5}{8}73​<85​.

You use cross multiplication here to compare unlike fractions without finding a common denominator explicitly.

2\. When You Do NOT Cross Multiply

This is where a lot of confusion comes from on forums and in class discussions.

You do NOT cross multiply when:

  • You are simply multiplying fractions like 23×45\frac{2}{3}\times \frac{4}{5}32​×54​.
    • Here you “multiply straight across”:
      • Tops: 2×4=82\times 4=82×4=8
      • Bottoms: 3×5=153\times 5=153×5=15
    • So 23×45=815\frac{2}{3}\times \frac{4}{5}=\frac{8}{15}32​×54​=158​.
  • You are adding or subtracting fractions.
    • For 13+14\frac{1}{3}+\frac{1}{4}31​+41​, you find a common denominator (like 12), you don’t cross multiply to solve the sum.
  • You have an expression with fractions but no “fraction = fraction” structure.
    • If there’s no equality between two simple fractions, cross multiplication usually isn’t the intended method.

A nice rule of thumb from teachers and math help communities:

“When you have fraction = fraction, you can cross multiply.
When you’re just multiplying fractions, multiply straight across.”

3\. Mini Cheat Sheet: When Do You Cross Multiply Fractions?

[3][7][9] [10][6][1] [7][9] [3]
Situation Example Do you cross multiply? What you do instead / extra note
Solving a proportion (fraction = fraction) $$\frac{2}{3} = \frac{x}{9}$$ Yes ✅ Cross multiply: $$2 \times 9 = 3 \times x$$, then solve for $$x$$.
Comparing two fractions $$\frac{3}{7}$$ vs $$\frac{5}{8}$$ Yes ✅ Cross multiply to compare 24 and 35; larger product ↔ larger fraction.
Multiplying fractions $$\frac{2}{3} \times \frac{4}{5}$$ No ❌ Multiply straight across: $$\frac{8}{15}$$.
Adding/subtracting fractions $$\frac{1}{3} + \frac{1}{4}$$ No ❌ Find a common denominator, then add numerators.
More complex equations with many terms $$\frac{x}{2} + \frac{3}{x+1} = \frac{3x - 5}{4x + 4}$$ Often no ❌ Can multiply through by common denominator; “pure” cross multiplication is mainly for one fraction = one fraction.

4\. A Tiny Story to Lock It In

Imagine two different “worlds” of fraction problems:
  • In the Proportion World , every problem looks like a scale: one fraction on the left, one on the right, perfectly balanced. That’s when cross multiplication is your favorite shortcut to keep the balance while getting rid of denominators.
  • In the Multiply World , fractions are teammates, not opponents: they stand side by side with a multiplication sign, so they just multiply straight across the top and straight across the bottom. No crossing lines needed.

Remember:

If it’s “fraction equals fraction” or “which fraction is bigger?”, cross multiply.
If it’s “fraction times fraction”, multiply straight across.

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TL;DR
You cross multiply fractions when:

  1. You have a proportion: one fraction equal to another;
  2. You are comparing which of two fractions is larger.
    You do not cross multiply when simply multiplying or adding fractions.

Information gathered from public forums or data available on the internet and portrayed here.