Short Answer
Cross-multiplication in proportion is a method used to compare two ratios or find a missing value. It is based on the rule that in a proportion a:b = c:d, we multiply across the numbers to get a × d = b × c.
This method helps us check if two ratios are equal or solve equations easily. It is simple and widely used in mathematics for solving proportion problems.
Detailed Explanation:
Cross multiplication in proportion
Cross multiplication is a simple and important method used in proportions. A proportion is written as a:b = c:d or a/b = c/d. In this method, we multiply the first term with the last term and the second term with the third term. This gives us the equation a × d = b × c.
Meaning of cross multiplication
Cross multiplication means multiplying numbers diagonally across the equality sign. For example, in the proportion 2:3 = 4:6, we multiply 2 with 6 and 3 with 4. Both give the same result, which is 12. This shows that the two ratios are equal.
Rule of cross multiplication
The rule of cross multiplication states that if two ratios are equal, then the product of their cross terms will also be equal. This rule is based on the fundamental rule of proportion. It helps us convert a proportion into a simple multiplication equation.
Checking correctness of proportion
Cross multiplication helps us check whether a proportion is correct. For example, in 3:5 = 9:15, we multiply crosswise. 3 × 15 = 45 and 5 × 9 = 45. Since both sides are equal, the proportion is correct. If the products are not equal, then the proportion is wrong.
Finding missing value
One of the most important uses of cross multiplication is finding unknown values. For example, if 4:7 = x:14, we apply cross multiplication. Multiply 4 × 14 = 7 × x, which gives 56 = 7x. Solving this, we get x = 8. This method makes solving equations very easy.
Steps of cross multiplication
First, write the proportion in the form a:b = c:d. Second, multiply the first and last terms. Third, multiply the middle terms. Fourth, equate both products and solve the equation if needed. Following these steps ensures correct results.
Importance in mathematics
Cross multiplication is very important because it simplifies problems involving proportions. It helps in solving equations quickly and accurately. It is widely used in algebra, arithmetic, and other areas of mathematics.
Use in daily life
Cross multiplication is useful in many real-life situations. It is used in scaling recipes, calculating speed, and solving problems related to cost and quantity. For example, if the price of 2 items is known, we can find the price of more items using this method.
Common mistakes
A common mistake is multiplying wrong pairs of numbers. The multiplication must be done diagonally, not straight across. Another mistake is writing the proportion incorrectly before applying the method. Careful steps should be followed to avoid errors.
Conclusion
Cross multiplication is a simple method used in proportion to check equality and find unknown values. It is based on multiplying terms diagonally and equating the results. It is very useful in mathematics and everyday problem-solving.