Problem 11 - Elementary Algebra

Question

When the same number is subtracted from both the numerator and denominator of \(\frac{1}{10},\) the result is \(\frac{2}{3}\). What is the number?

Step-by-Step Solution

Answer

Answer: -17

Step 1: Write the equation

Write the equation by subtracting x from both the numerator and denomoninator of the fraction \(\frac{1}{10}\). This should result in the fraction \(\frac{2}{3}\). So, we have: $$\frac{1-x}{10-x}=\frac{2}{3}$$

Step 2: Cross-multiply the fractions

To solve the equation, cross-multiply the fractions: $$3(1-x) = 2(10-x)$$

Step 3: Distribute the factors

Distribute the 3 and 2 on both sides of the equation: $$3-3x = 20-2x$$

Step 4: Isolate the variable x

Subtract 20 from both sides and add 3x to both sides to isolate x: $$3-20 = -2x + 3x$$

Step 5: Solve for x

Simplify the equation and solve for the value of x: $$-17=x$$ The number is -17. When -17 is subtracted from both the numerator and denominator of the fraction \(\frac{1}{10}\), the result is \(\frac{2}{3}\).

Key Concepts

Cross-Multiplication

Cross-multiplication is a technique used to solve equations that involve fractions. It involves multiplying the numerator of one fraction by the denominator of the other fraction and vice versa, to eliminate the fractions and create a simpler equation. For example, given the equation \(\frac{a}{b} = \frac{c}{d}\), cross-multiplication would give us \(ad = bc\).

This powerful tool is particularly useful when dealing with proportions or equations like the one in our exercise \(\frac{1-x}{10-x} = \frac{2}{3}\). By cross-multiplying, we obtain \(3(1-x) = 2(10-x)\), which simplifies the equation and allows us to work with integers rather than fractions, making the subsequent steps more straightforward.

Isolating Variables

Isolating a variable means rearranging an equation so that one variable stands alone on one side of the equality sign. This is a fundamental step in solving equations because it helps us find the value of that variable. We do this by performing inverse operations on both sides of the equation to maintain balance.

In our example, after cross-multiplication, we arrive at an equation \(3-3x = 20-2x\). To isolate \(x\), we need to get all terms with \(x\) on one side and the constant terms on the other. So, we add \(2x\) to both sides and subtract \(20\) from both sides, resulting in \(3-20 = -2x + 3x\) which simplifies to \(x = -17\). Isolating variables helps us to pinpoint the exact value needed to satisfy the original problem.

Distributive Property

The distributive property is a rule in algebra that allows us to remove parentheses by expanding expressions. It states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products. For example, \(a(b + c) = ab + ac\).

In step 3 of our problem, we distributed the numbers \(3\) and \(2\) across the parentheses, transforming \(3(1-x)\) into \(3-3x\) and \(2(10-x)\) into \(20-2x\). This property not only simplifies equations but ensures every term is accounted for before further steps, such as isolating the variable, are taken. Understanding the distributive property is essential for effectively dealing with algebraic expressions and equations.