Cross Multiplication Method
Cross multiplication is a technique for solving equations where two fractions are set equal to each other. It is one of the most useful shortcuts in algebra.
When you have an equation of the form a/b = c/d, cross multiplication gives you ad = bc. This removes the fractions completely, leaving a simpler equation to solve.
This method is used extensively in solving proportions, fractional equations, and ratio problems in Class 8 and beyond.
What is Cross Multiplication Method?
Definition: Cross multiplication is the process of multiplying the numerator of one fraction by the denominator of the other fraction, and setting the two products equal.
If a/b = c/d, then a × d = b × c
Where:
- a = numerator of the left fraction
- b = denominator of the left fraction
- c = numerator of the right fraction
- d = denominator of the right fraction
- b ≠ 0 and d ≠ 0
Why it works:
- If a/b = c/d, multiply both sides by bd.
- LHS: (a/b) × bd = ad
- RHS: (c/d) × bd = bc
- Result: ad = bc
Methods
Steps to solve using cross multiplication:
- Write the equation in the form: (expression 1) / (expression 2) = (expression 3) / (expression 4).
- Cross multiply: Multiply the numerator of the left side by the denominator of the right side, and the numerator of the right side by the denominator of the left side.
- Set them equal: numerator₁ × denominator₂ = numerator₂ × denominator₁.
- Expand both sides (if they contain brackets).
- Transpose variable terms to one side and constants to the other.
- Solve for the variable.
- Verify by substituting back into the original equation.
When to use cross multiplication:
- When the equation has the form: one fraction = one fraction.
- When solving proportions (a : b = c : d).
- When comparing ratios.
When NOT to use cross multiplication:
- When there are more than two fractions in the equation (use LCM method instead).
- When fractions are added or subtracted (e.g., x/2 + x/3 = 5 — use LCM method).
Solved Examples
Example 1: Example 1: Simple proportion
Problem: Solve: x/5 = 6/10
Solution:
Steps:
- Cross multiply: x × 10 = 5 × 6
- 10x = 30
- x = 3
Check: LHS = 3/5 = 0.6. RHS = 6/10 = 0.6. Correct.
Answer: x = 3
Example 2: Example 2: Variable in numerator
Problem: Solve: (2x + 1)/3 = 5/3
Solution:
Steps:
- Cross multiply: 3(2x + 1) = 3 × 5
- 6x + 3 = 15
- 6x = 12
- x = 2
Check: LHS = (2(2) + 1)/3 = 5/3. RHS = 5/3. Correct.
Answer: x = 2
Example 3: Example 3: Variables on both sides
Problem: Solve: (3x − 2)/4 = (2x + 1)/3
Solution:
Steps:
- Cross multiply: 3(3x − 2) = 4(2x + 1)
- Expand: 9x − 6 = 8x + 4
- Transpose: 9x − 8x = 4 + 6
- x = 10
Check: LHS = (30 − 2)/4 = 28/4 = 7. RHS = (20 + 1)/3 = 21/3 = 7. Correct.
Answer: x = 10
Example 4: Example 4: Negative coefficients
Problem: Solve: (5x + 3)/7 = (x − 9)/2
Solution:
Steps:
- Cross multiply: 2(5x + 3) = 7(x − 9)
- Expand: 10x + 6 = 7x − 63
- Transpose: 10x − 7x = −63 − 6
- 3x = −69
- x = −23
Check: LHS = (5(−23) + 3)/7 = (−115 + 3)/7 = −112/7 = −16. RHS = (−23 − 9)/2 = −32/2 = −16. Correct.
Answer: x = −23
Example 5: Example 5: Variable in denominator
Problem: Solve: 4/(x + 2) = 2/5
Solution:
Steps:
- Cross multiply: 4 × 5 = 2 × (x + 2)
- 20 = 2x + 4
- Transpose: 2x = 20 − 4
- 2x = 16
- x = 8
Check: LHS = 4/(8 + 2) = 4/10 = 2/5. RHS = 2/5. Correct.
Answer: x = 8
Example 6: Example 6: Proportion word problem
Problem: If 3 : x = 5 : 15, find x.
Solution:
Given:
- 3/x = 5/15
Steps:
- Cross multiply: 3 × 15 = 5 × x
- 45 = 5x
- x = 9
Check: 3/9 = 1/3. 5/15 = 1/3. Correct.
Answer: x = 9
Example 7: Example 7: Both numerators have expressions
Problem: Solve: (4x − 5)/(2x + 3) = 3/5
Solution:
Steps:
- Cross multiply: 5(4x − 5) = 3(2x + 3)
- Expand: 20x − 25 = 6x + 9
- Transpose: 20x − 6x = 9 + 25
- 14x = 34
- x = 34/14 = 17/7
Check: LHS numerator = 4(17/7) − 5 = 68/7 − 35/7 = 33/7. LHS denominator = 2(17/7) + 3 = 34/7 + 21/7 = 55/7. LHS = (33/7)/(55/7) = 33/55 = 3/5. RHS = 3/5. Correct.
Answer: x = 17/7
Example 8: Example 8: Finding the ratio
Problem: The ratio of (2x + 1) to (3x − 1) is 3 to 4. Find x.
Solution:
Given:
- (2x + 1)/(3x − 1) = 3/4
Steps:
- Cross multiply: 4(2x + 1) = 3(3x − 1)
- Expand: 8x + 4 = 9x − 3
- Transpose: 8x − 9x = −3 − 4
- −x = −7
- x = 7
Check: (2(7) + 1)/(3(7) − 1) = 15/20 = 3/4. Correct.
Answer: x = 7
Example 9: Example 9: Word problem — ages in ratio
Problem: The ages of A and B are in the ratio 3 : 5. After 6 years, their ages will be in the ratio 2 : 3. Find their present ages.
Solution:
Given:
- Present ages: A = 3x, B = 5x
- After 6 years: A = 3x + 6, B = 5x + 6
Equation: (3x + 6)/(5x + 6) = 2/3
- Cross multiply: 3(3x + 6) = 2(5x + 6)
- 9x + 18 = 10x + 12
- 9x − 10x = 12 − 18
- −x = −6
- x = 6
Answer: A's age = 3(6) = 18 years. B's age = 5(6) = 30 years.
Example 10: Example 10: Converting to fraction = fraction form
Problem: Solve: (x + 3)/6 − 1 = x/4
Solution:
Steps:
- Rewrite: (x + 3)/6 − 1 = x/4
- Move −1 to RHS: (x + 3)/6 = x/4 + 1
- Write RHS as single fraction: x/4 + 1 = (x + 4)/4
- Now: (x + 3)/6 = (x + 4)/4
- Cross multiply: 4(x + 3) = 6(x + 4)
- 4x + 12 = 6x + 24
- 4x − 6x = 24 − 12
- −2x = 12
- x = −6
Check: LHS = (−6 + 3)/6 − 1 = (−3)/6 − 1 = −1/2 − 1 = −3/2. RHS = −6/4 = −3/2. Correct.
Answer: x = −6
Real-World Applications
Real-world applications of cross multiplication:
- Proportions: If 5 books cost Rs 200, how much do 8 books cost? Set up 5/200 = 8/x and cross multiply.
- Map scales: If 1 cm on a map represents 5 km, and two cities are 3.5 cm apart, find actual distance using proportions.
- Recipe scaling: If a recipe for 4 people needs 3 cups of flour, how much for 10 people? 4/3 = 10/x.
- Currency conversion: If 1 USD = 83 INR, how many dollars is 4150 INR? Set up the proportion and cross multiply.
- Speed-time calculations: Using d = s × t in fraction form.
- Ratio problems: Finding unknowns when ratios are given.
Key Points to Remember
- Cross multiplication applies when the equation is in the form a/b = c/d.
- Cross multiplying gives ad = bc, eliminating all fractions.
- This method works because multiplying both sides by bd preserves equality.
- Use cross multiplication for proportions and equations with one fraction on each side.
- Do NOT use cross multiplication when fractions are added or subtracted (use LCM method instead).
- After cross multiplying, expand brackets and solve by transposition.
- Always check that the solution does not make any denominator zero.
- Cross multiplication is equivalent to finding the fourth proportional.
- The method extends to ratios: if a : b = c : d, then ad = bc.
- Always verify the answer by substituting back into the original equation.
Practice Problems
- Solve: x/7 = 12/21
- Solve: (x + 5)/3 = (2x − 1)/4
- Solve: (3x + 4)/5 = (x + 8)/3
- Solve: 7/(x − 1) = 3/4
- Solve: (2x − 3)/(x + 1) = 4/5
- If 4 : 7 = x : 35, find x.
- The ratio of two numbers is 5 : 8. Their sum is 91. Find the numbers.
- Solve: (5x + 1)/(3x − 2) = 7/4
Frequently Asked Questions
Q1. What is cross multiplication?
Cross multiplication is a method to solve equations of the form a/b = c/d by multiplying across: ad = bc. It eliminates fractions and gives a simpler equation.
Q2. When can I use cross multiplication?
Use it when the equation has exactly one fraction on the left side equal to one fraction on the right side. It does not work directly when fractions are added or subtracted.
Q3. Why is it called 'cross' multiplication?
Because you multiply diagonally across the equals sign: the numerator of the left with the denominator of the right, and vice versa. The multiplication paths form an X (cross) shape.
Q4. Can I cross multiply if the variable is in the denominator?
Yes. For example, in 5/(x + 2) = 3/7, cross multiply to get 5 × 7 = 3(x + 2), which gives 35 = 3x + 6, so x = 29/3. Just verify the answer does not make the denominator zero.
Q5. What if both numerator and denominator have the variable?
Cross multiply as usual. For example, (2x + 1)/(x − 3) = 4/5 gives 5(2x + 1) = 4(x − 3). Expand and solve. Check that x ≠ 3.
Q6. Is cross multiplication the same as solving a proportion?
Yes. A proportion is a statement that two ratios are equal (a/b = c/d). Cross multiplication is the standard method to solve proportions.
Q7. What mistakes should I avoid?
Common mistakes: (1) Trying to cross multiply when fractions are added (use LCM instead). (2) Forgetting to expand brackets after cross multiplying. (3) Forgetting to check if the denominator becomes zero.
Q8. Can I cross multiply with more than two fractions?
No. Cross multiplication only works with exactly two fractions set equal. For more than two fractions or sums of fractions, use the LCM method.
Q9. What is the relationship between cross multiplication and equivalent fractions?
Two fractions a/b and c/d are equivalent if and only if ad = bc. Cross multiplication checks this equality.
Q10. How is cross multiplication used in ratios?
If a : b = c : d (meaning a/b = c/d), then ad = bc. This is called the 'product of extremes equals product of means' rule. It is the same as cross multiplication.
