Problem 60 - Elementary Algebra

Question

Translate the phrases or sentences to mathematical expressions or equations. One seventh of a number plus two ninths of the number.

Step-by-Step Solution

Answer

Question: Translate the sentence "the sum of one seventh and two ninths of a number" into a mathematical expression. Answer: \(\frac{23}{63}x\)

Step 1: Represent the given number as a variable

Let the given number be x.

Step 2: Write fractions for 'one seventh of x' and 'two ninths of x'

One seventh of x can be written as \(\frac{1}{7}x\), and two ninths of x can be written as \(\frac{2}{9}x\).

Step 3: Write the expression for the sum of the two fractions

The sum of \(\frac{1}{7}x\) and \(\frac{2}{9}x\) can be written as \(\frac{1}{7}x + \frac{2}{9}x\).

Step 4: Find the least common multiple (LCM) of the denominators

To add the fractions, we need a common denominator. The LCM of 7 and 9 is 63.

Step 5: Convert the given fractions into equivalent fractions with the LCM as their denominator and add the fractions

To get the common denominator, we need to multiply and divide each fraction by the same number such that the denominator becomes 63. For the first fraction, we multiply both the numerator and denominator by 9; for the second fraction, we multiply both the numerator and the denominator by 7. This gives us the equivalent fractions: \(\frac{9}{63}x\) and \(\frac{14}{63}x\). The sum of these fractions is: \(\frac{9}{63}x + \frac{14}{63}x = \frac{9 + 14}{63}x\)

Step 6: Simplify the expression

Simplify the expression by adding the numerators: \(\frac{23}{63}x\). The final expression representing the given problem in mathematical language is: \(\frac{23}{63}x\).

Key Concepts

Fractions

Fractions are parts of a whole. They consist of two main components: the numerator and the denominator.
- The **numerator** is the top number of a fraction and represents how many parts of the whole are being considered.
- The **denominator** is the bottom number and shows the total number of equal parts the whole is divided into.

In the expression \(rac{1}{7}x\), '1' is the numerator and '7' is the denominator. This indicates that the variable \(x\) is divided into 7 equal parts, and we are taking 1 part of it.
Similarly, \(rac{2}{9}x\) suggests taking 2 parts out of 9 equal parts of \(x\). Understanding fractions is crucial, especially in algebra, where they are used to formulate expressions involving parts of a number. Knowing how to write and manipulate fractions allows us to describe relationships and solve equations accurately.

Common Denominators

To add fractions, they need to share a common denominator. This means the fractions are expressed over the same base (in the denominator) before they can be directly added or subtracted.
For example, when dealing with \(\frac{1}{7}x\) and \(\frac{2}{9}x\), the denominators are 7 and 9. To add them, we find the least common multiple (LCM) of 7 and 9, which is 63.
We then convert each fraction to an equivalent fraction with a denominator of 63:

• Multiply the numerator and denominator of \(\frac{1}{7}x\) by 9 to get \(\frac{9}{63}x\).
• Multiply the numerator and denominator of \(\frac{2}{9}x\) by 7 to get \(\frac{14}{63}x\).

This way, both fractions share a common denominator of 63, making it simpler to perform addition.

Variables

Variables are symbols used to represent unknown numbers or values in mathematical expressions and equations. They allow us to create general formulas that hold true for different numbers.
In our problem, the variable is \(x\), representing the unknown number we are working with. This usage of variables is fundamental in algebra since they enable us to describe relationships and compute various scenarios without specifying the exact numbers initially.
This characteristic of variables is what makes algebra a powerful tool for solving real-world problems by providing a way to model different situations efficiently.

Numerical Coefficients

Numerical coefficients are the specific numbers that multiply the variables in an algebraic expression. They give the terms their particular value in the context of the expression or equation.
In \(\frac{1}{7}x\), the numerical coefficient is \(\frac{1}{7}\), representing how much of \(x\) is being considered. Similarly, in \(\frac{2}{9}x\), \(\frac{2}{9}\) is the coefficient.
These coefficients tell us how much of the whole (represented by \(x\)) is taken in each part of the expression.
Understanding and manipulating numerical coefficients are important skills in algebra, as they help determine the structure and outcomes of equations and expressions.