Problem 25 - Fundamentals of Mathematics

Question

\(\frac{3}{14}\) of what number is \(\frac{6}{7} ?\)

Step-by-Step Solution

Answer

The number is 4.

Step 1: Set Up the Equation

To find a number (let's call it \( x \)) such that \( \frac{3}{14} \times x = \frac{6}{7} \), we'll start by setting up an equation based on the description.

Step 2: Simplify the Equation

Rewrite the initial equation as \( \frac{3}{14} \times x = \frac{6}{7} \). Our goal is to isolate \( x \) and solve for it.

Step 3: Solve for x

Multiply both sides of the equation by \( \frac{14}{3} \) (which is the reciprocal of \( \frac{3}{14} \)). This gives us \( x = \frac{6}{7} \times \frac{14}{3} \).

Step 4: Simplify the Right Side

Calculate \( \frac{6}{7} \times \frac{14}{3} \). This multiplication involves multiplying the numerators together and the denominators together: \( \frac{6 \times 14}{7 \times 3} = \frac{84}{21} \).

Step 5: Final Simplification

Simplify \( \frac{84}{21} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 21. This results in \( 4 \).

Key Concepts

Equation Solving

Equation solving often involves finding the value of an unknown variable in an equation. To tackle these problems, it's crucial to understand the structure of the equation. In our example,

• We have the equation \(\frac{3}{14} \times x = \frac{6}{7}\).
• The target is to isolate \(x\) and solve for it.

To achieve this, you must perform operations that allow you to "undo" what's being done to \(x\). This is similar to peeling off layers of an onion to reach the core. By multiplying both sides by the reciprocal of the fraction connected to \(x\), you isolate \(x\) and reveal its value.
This key method works because you are essentially multiplying by \(1\), cleverly disguised as \(\frac{14}{3} \times \frac{3}{14}\). This operation counteracts the initial fraction multiplication, letting you solve the equation.

Multiplication of Fractions

When you multiply fractions, the rule is to multiply the numerators together and the denominators together. Let's take a look:

• The general formula is: \(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\).
• For our example, \(\frac{6}{7} \times \frac{14}{3} = \frac{6 \times 14}{7 \times 3}\).

This gives \(\frac{84}{21}\). By focusing on this simple multiplication, you ensure that each part of the fraction is correctly combined.
Be sure to multiply the top numbers (numerators) and the bottom numbers (denominators) separately, keeping each aspect in its rightful place.
This careful organization is what makes fraction multiplication both straightforward and reliable.

Simplification in Mathematics

Simplification is all about reducing expressions to their simplest form. For our answer, \(\frac{84}{21}\), we need to simplify:

• Identify the greatest common divisor (GCD) of the numbers.
• Both 84 and 21 are divisible by 21.

When you divide the numerator and denominator by 21, you get \(\frac{84 \div 21}{21 \div 21} = 4\). This process of simplification is essential because it makes the result cleaner and more usable.
Always try to reduce fractions to their simplest form because it is easier to understand and clearly communicates the actual quantity.
To effectively simplify, look for the largest number that divides both the numerator and denominator without leaving a remainder. This makes sure your simplified fraction maintains its original value now, just clearer!