Problem 51 - Elementary Algebra

Question

If the numerator of a certain fraction is increased by 5 . and the denominator is decreased by 1, the resulting fraction is \(\frac{8}{3}\). However, if the numerator of the original fraction is doubled, and the denominator of the original fraction is increased by 7 , the resulting fraction is \(\frac{6}{11}\). Find the original fraction.

Step-by-Step Solution

Answer

The original fraction is \(\frac{4}{7}\).

Step 1: Set Up the Variables

Let the original fraction be \( \frac{x}{y} \). Our goal is to find the values of \( x \) (numerator) and \( y \) (denominator) that satisfy the given conditions.

Step 2: Translate the First Condition

According to the problem, if the numerator is increased by 5 and the denominator is decreased by 1, the fraction becomes \( \frac{8}{3} \). Thus, the equation is: \[ \frac{x + 5}{y - 1} = \frac{8}{3} \].

Step 3: Translate the Second Condition

Next, if the numerator is doubled and the denominator is increased by 7, the fraction becomes \( \frac{6}{11} \). Thus, the equation is: \[ \frac{2x}{y + 7} = \frac{6}{11} \].

Step 4: Solve the First Equation

Cross-multiply in the first equation to eliminate the fraction: \(3(x + 5) = 8(y - 1)\). Simplify to get \(3x + 15 = 8y - 8\). Rearrange to form the equation: \(3x - 8y = -23\).

Step 5: Solve the Second Equation

Cross-multiply in the second equation to eliminate the fraction: \(11(2x) = 6(y + 7)\). Simplify to get \(22x = 6y + 42\). Rearrange to form the equation: \(22x - 6y = 42\).

Step 6: Simplify the System of Equations

From the two equations, we have: 1. \(3x - 8y = -23\)2. \(22x - 6y = 42\).

Step 7: Solve the System Using Substitution or Elimination

Let's use elimination. Multiply the first equation by 2: \(6x - 16y = -46\) and the second equation by 3: \(66x - 18y = 126\). Subtract the first from the second to eliminate \(y\):\(66x - 18y - (6x - 16y) = 126 - (-46)\)\(60x - 2y = 172\).Now solve \(18x - 16y = 92\).Re-substitute this back to solve for \(y\) .

Step 8: Find the Values of x and y

Solving the equations gives us \(x = 4\) and \(y = 7\). Therefore, the original fraction is \(\frac{4}{7}\).

Key Concepts

Fraction Simplification

Fraction simplification means reducing a fraction to its simplest form. This involves dividing both the numerator and the denominator by their greatest common divisor (GCD). Simplifying fractions makes them easier to understand and work with.

For instance, consider the fraction \( \frac{12}{8} \). Both 12 and 8 can be divided by 4, which is their GCD, giving us \( \frac{3}{2} \). This is a simpler version of the original fraction.

By simplifying fractions, you ensure they are in the simplest form, which is useful for comparing, adding, subtracting, or integrating them into more complex equations. The problem-solving exercise involving fraction simplification commonly uses this principle to ease calculations.

System of Equations

A system of equations consists of two or more equations that have common variables. The goal is to find values for these variables that satisfy all equations at the same time. A real-life example includes determining grades based on weighted categories or finding equilibrium in market supply and demand.

In our exercise, we worked with a system consisting of the equations: \(3x - 8y = -23\) and \(22x - 6y = 42\). These equations are based on the conditions provided in the problem statement.

To solve such systems, you can use methods like substitution (solving one equation for a variable and substituting it into the other) or elimination (adding or subtracting equations to eliminate a variable). In this exercise, elimination was applied by aligning the coefficients of one variable, thus simplifying the calculation process. Solving a system of equations is a crucial algebra skill allowing you to find precise values for variables.

Cross Multiplication

Cross multiplication is a technique used to solve equations involving fractions. It is especially useful when equations are proportionate and can be expressed in the form \( \frac{a}{b} = \frac{c}{d} \). Cross multiplication eliminates the fractions, allowing us to solve for the unknowns within the proportions more easily.

When using cross multiplication, you multiply the numerator of one fraction by the denominator of the other and vice versa. This results in the equation \( a \times d = b \times c \).

In the original exercise, cross multiplication was applied in steps when simplifying both conditions: reducing \( \frac{x + 5}{y - 1} = \frac{8}{3} \) to \( 3(x + 5) = 8(y - 1) \) and \( \frac{2x}{y + 7} = \frac{6}{11} \) to \( 11(2x) = 6(y + 7) \). This made it easier to handle the equations in a systematic way, key to algebraic problem-solving.