Problem 29 - Algebra 1

Question

\(\frac{5}{8}-p\) when \(p=\frac{3}{16}\)

Step-by-Step Solution

Answer

The solution to the exercise is \(\frac{7}{16}\).

Step 1: Substitute the value of p

Place the known value of \(p\), which is \(\frac{3}{16}\), into the equation, resulting in the following: \(\frac{5}{8} - \frac{3}{16}\).

Step 2: Make the denominators the same

To subtract these fractions, the denominators must be the same. Multiply the first fraction by \(\frac{2}{2}\) to match the fractions: \(=\frac{10}{16} - \frac{3}{16}\).

Step 3: Subtract the fractions

Subtract the second fraction from the first one now that the denominators are the same: \(\frac{10}{16} - \frac{3}{16}=\frac{7}{16}\).

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. They are used to represent relationships, patterns, and general rules within mathematics. In the given exercise, \(\frac{5}{8}-p\) is an algebraic expression where \(p\) is a variable that can take on different values. Simplifying this expression involves substituting the variable \(p\) with its known value and performing arithmetic operations to find a result.

When substituting a value into an algebraic expression, make sure to replace every instance of the variable with the given number or expression, as was done in the exercise by replacing \(p\) with \(\frac{3}{16}\). This substitution transforms our algebraic expression into a numerical one, which can then be calculated using arithmetic operations.

Equivalent Fractions

Equivalent fractions are different fractions that represent the same part of a whole. They are key when working with fractions because they allow us to add, subtract, multiply, or divide fractions with different denominators. The concept of equivalent fractions was used in Step 2 of the provided solution where the fraction \(\frac{5}{8}\) was rewritten as \(\frac{10}{16}\) by multiplying both its numerator and denominator by 2.

By finding equivalent fractions that have a common denominator, the subtraction of fractions becomes simpler. This is because once the denominators match, you only need to subtract the numerators to find the answer. Remember that the denominators must stay the same after the subtraction.

Simplifying Fractions

Simplifying fractions, also known as reducing fractions, is the process of making a fraction as simple as possible. This is commonly done by finding the greatest common divisor (GCD) of the numerator and the denominator and dividing them by this number. However, in the final answer of our exercise, \(\frac{7}{16}\), no further simplification was required because 7 and 16 have no common factors other than 1.

In general, if the numerator and denominator of a fraction cannot be divided by the same number to produce whole numbers, the fraction is already in simplest form. Simplifying fractions is a crucial step in ensuring clarity and accuracy in mathematical communication and is a habit that should be developed for efficient mathematical problem-solving.