Problem 66 - Prealgebra

Question

Use the digits 3, 4, 5, 6, 8, or 9 no more than once to make true sentences. $$\frac{\square}{\square} \times \frac{\square}{\square}=\frac{5}{8}$$

Step-by-Step Solution

Answer

Combine \( \frac{5}{6} \) and \( \frac{3}{4} \) to satisfy \( \frac{5}{8} \).

Step 1: Understanding the Equation

We are given the equation \( \frac{\square}{\square} \times \frac{\square}{\square} = \frac{5}{8} \) and have to find a combination of the digits 3, 4, 5, 6, 8, or 9, using each digit at most once, to satisfy the equation.

Step 2: Considering Possible Pairings

The fractions \( \frac{a}{b} \) and \( \frac{c}{d} \) must satisfy the condition \( \frac{a \times c}{b \times d} = \frac{5}{8} \). Let's consider suitable values for \( a \times c \) and \( b \times d \) which must simplify to 5 and 8, respectively.

Step 3: Matching Numerator Product

One possible match for \( a \times c = 5 \) is if either \( a = 5 \) and \( c = 1 \) or \( a = 1 \) and \( c = 5 \). However, the digit 1 is not available, so we reconsider possible products from the given digits.

Step 4: Construct Numerators and Denominators

We try potential numerators as \( a = 5 \) and \( c = 9 \). The product becomes \( 5 \times 9 = 45 \), which is not a possible simplification to 5 using allowed numbers. Testing combinations, use \( a = 5 \) and \( c = 4 \) to get \( a \times c = 20 \), then find corresponding denominators \( b \) and \( d \) so that the product can simplify to 8.

Step 5: Confirm Valid Pairing

Let's use \( \frac{5}{6} \) and \( \frac{4}{3} \). Then we verify: \[ \frac{5 \times 4}{6 \times 3} = \frac{20}{18} \]. This needs further simplification, thus reconsider numerators or possible denominators.

Step 6: Solving Using Appropriate Values

Using \( \frac{5}{9} \times \frac{6}{4} = \frac{30}{36} \). Simplifying \( \frac{30}{36} \) gives \( \frac{5}{6} \), not matching \( \frac{5}{8} \). Readjust choices for valid answer. Try \( \frac{3}{6} \times \frac{5}{4} = \frac{15}{24} \), not correct. Another attempt: \( \frac{6}{9} \times \frac{5}{4} = \frac{30}{36} \). Still incorrect.

Step 7: Verify Correct Combination

Use \( \frac{5}{9} \times \frac{6}{4} \): this simplifies \( \frac{30}{36} = \frac{5}{6} \). Re-evaluate fraction and selections for verifying assumptions. Correct sequence \( \frac{5}{8} = \frac{5}{8} \, (\text{original target}) \): recalibrate assumptions and verify anew.

Step 8: Final Verification

The viable combination to examine needed is \( \frac{5}{6} \times \frac{3}{4} \). Performing multiplication yields \( \frac{15}{24} \). Simplify: \( \frac{15}{24} = \frac{5}{8} \). This gives the correct estimation and satisfies the given equation.

Key Concepts

Fractions Multiplication

Multiplying fractions involves a simple but specific process. To multiply two fractions, you must multiply the numerators together to obtain a new numerator, and multiply the denominators together to obtain a new denominator. For example, if you have

• two fractions: \( \frac{a}{b} \) and \( \frac{c}{d} \),
• multiply the numerators: \( a \times c \) and the denominators: \( b \times d \).

This will give you a new fraction: \( \frac{a \times c}{b \times d} \). It is crucial to understand that the multiplication is straight across the fraction bars. There is no need for a common denominator. This simplicity is one reason why multiplying fractions is often preferred in fraction operations.
For instance, when the exercise asks to use digits (3, 4, 5, 6, 8, 9) to fit into \( \frac{\square}{\square} \times \frac{\square}{\square} = \frac{5}{8} \), understanding this rule helps in trial and error for finding the suitable numerators and denominators.

Simplifying Fractions

Once you have multiplied your fractions and obtained a result, the next important step is simplifying the fraction. Simplification means reducing the fraction to its simplest form where the numerator and denominator are as small as possible and they no longer share any common factors other than 1. Simplifying ensures that your answer is presented neatly and is universally understood.
To simplify a fraction like \( \frac{15}{24} \), consider the steps:

• Find the greatest common divisor (GCD) of the numerator and denominator, which is the largest number that divides both without a remainder.
• In this case, the GCD of 15 and 24 is 3.
• Divide both numerator and denominator by their GCD: \( \frac{15 \div 3}{24 \div 3} = \frac{5}{8} \).

Simplification is critical in getting the final solution in the exercise. After computing \( \frac{15}{24} \), simplifying it to \( \frac{5}{8} \) confirms it matches the target fraction from the equation.

Numerator and Denominator

Understanding what numerators and denominators are will help you in managing fractions more effectively. In any fraction \( \frac{a}{b} \):

• The numerator \( a \) is the number above the fraction bar. It indicates how many parts of a whole you have.
• The denominator \( b \) is the number below the fraction bar. It indicates into how many parts the whole is divided.

For example, in the fraction \( \frac{5}{8} \), 5 is the numerator, and 8 is the denominator. Numerators and denominators play pivotal roles in various operations with fractions, including the one given in the exercise. It becomes especially crucial when determining possible products and sums from a set of numbers, as you need a keen awareness of how these numbers play their roles within a fraction.
When you try to solve for fractions in combinations, as in the exercise, considering both the suitability of the numerators and denominators and how they multiply is vital to finding the right solution that simplifies to the target fraction.