Solving Proportions by Cross-Multiplication: Guided Worksheet Plan

Algebra and Function Practice

A proportion states that two ratios are equal, such as 3/4 = x/12. Cross-multiplication turns this into a simple equation: multiply the numerator of one ratio by the denominator of the other. This lesson explains the method, shows a complete worked example, lists frequent mistakes, and provides four self-contained practice exercises with answers, reasoning, and check methods. All arithmetic uses consistent units and simple checkable values. The planned worksheet and answer key are described in the resource plan; they are not yet built.

Practice materials: Printable downloads are not available for this lesson yet. Use the instructions and four exercises below.

What Cross-Multiplication Does

In a proportion a/b = c/d, the cross products are a times d and b times c. Setting them equal gives ad = bc. This works because multiplying both sides of the proportion by bd clears the denominators. The result is a linear equation that can be solved for the missing term.

Cross-multiplication is valid only when the denominators are nonzero. Before solving, note any value that would make a denominator zero and exclude it. In this lesson all given denominators are positive, so no exclusion is needed beyond checking the final answer.

1. Write the proportion clearly, identifying the missing term.
2. Multiply the numerator of the left ratio by the denominator of the right ratio.
3. Multiply the denominator of the left ratio by the numerator of the right ratio.
4. Set the two products equal to form an equation.
5. Solve the equation for the missing term.
6. Substitute the result back into the original proportion to check.

Solving for a Missing Term

When the missing term is a numerator, the equation often has the form a/b = x/d. Cross-multiplication gives ad = bx, so x = ad/b. When the missing term is a denominator, as in a/b = c/x, cross-multiplication gives ax = bc, so x = bc/a. The same two products are formed either way; only the division step changes.

Keep units consistent. If the ratios compare quantities with units, the units must match on each side of the proportion. In pure-number exercises, treat all values as unitless. Always simplify fractions when possible, but keep the exact value for checking.

1. For a/b = x/d, compute x = (a times d) divided by b.
2. For a/b = c/x, compute x = (b times c) divided by a.
3. Reduce the result if it is a fraction.
4. Check by substituting and confirming both cross products are equal.

Checking Your Answer

A reliable check is to substitute the value back into the original proportion and compare the two cross products. If they are equal, the solution is correct. Another check is to simplify both ratios to lowest terms and confirm they match.

For decimal or fractional answers, keep enough precision to compare products exactly when possible. If rounding is used, state the rounding and confirm the products are close within the stated precision. In this lesson all answers are exact integers or simple fractions.

1. Substitute the answer into the original proportion.
2. Compute both cross products.
3. Confirm the products are equal.
4. Optionally simplify both ratios and confirm they are identical.

Worked example

Solve 6/8 = x/20 for x.

Four practice exercises

Common mistakes

• Multiplying numerator by numerator and denominator by denominator instead of cross-multiplying. Cross-multiplication pairs a numerator with the opposite denominator. Write the two products explicitly before equating them.
• Dividing by the wrong factor when isolating the missing term. After forming the equation, identify which side contains the missing term and divide both sides by its coefficient. Re-read the equation before dividing.
• Ignoring a denominator that could be zero. State the restriction before solving. If the missing term is a denominator, exclude zero from possible answers.
• Rounding intermediate values and losing exactness. Keep fractions exact until the final step. Round only if the task requires it, and state the precision.

Self-check rubric

• Identifies the missing term and writes the proportion clearly.
• Forms both cross products correctly, pairing each numerator with the opposite denominator.
• Sets the cross products equal and solves the resulting equation accurately.
• States any denominator restriction when relevant.
• Checks the answer by substitution or by simplifying both ratios.
• Presents work in a logical sequence with consistent units or unitless values.

Questions and answers