Solving for the Second Number in a Given Product and Fraction

Introduction

Solving for a missing number in a fraction product can be achieved through algebraic manipulation. This article will guide us through the process of finding the second number if we know the product and one of the numbers. We will explore the step-by-step solutions, apply algebraic rules, and verify our results for accuracy.

Detailed Explanation and Solution

Suppose we have two numbers, the product of which is (frac{2}{9}), and one of them is (frac{8}{21}). We need to find the other number. Let the other number be (x). Therefore, the equation becomes: [frac{8}{21} times x frac{2}{9}] To solve for (x), we isolate (x) by dividing both sides of the equation by (frac{8}{21}). This is equivalent to multiplying by the reciprocal of (frac{8}{21}), which is (frac{21}{8}). [begin{align*}x frac{2}{9} div frac{8}{21} x frac{2}{9} times frac{21}{8} x frac{2 times 21}{9 times 8} x frac{42}{72} x frac{7 times 6}{12 times 6} x frac{7}{12}end{align*}] Therefore, the other number is (frac{7}{12}).

Verification

Let's verify our solution by substituting (frac{7}{12}) back into the original product equation: [frac{8}{21} times frac{7}{12} frac{8 times 7}{21 times 12} frac{56}{252} frac{28}{126} frac{2}{9}] As we can see, the product is indeed (frac{2}{9}), confirming our solution is correct.

Conclusion

Finding the second number in a given product and fraction involves algebraic manipulation and utilization of fraction division rules. The key steps are to set up the equation, isolate the variable by multiplying by the reciprocal, simplify the fractions, and then verify the result. In this case, the other number is (frac{7}{12}), and it satisfies the condition that the product of (frac{8}{21}) and (frac{7}{12}) is (frac{2}{9}).

Related Keywords

Fraction product, dividing fractions, algebraic solution