Problem 51 - Elementary Algebra

Question

For the following problems, translate the following phrases or sentences into mathematical expressions or equations. A number is added to six and that result is multiplied by thirteen. This result is then divided by six times the number. The entire result is equal to fifty-nine.

Step-by-Step Solution

Answer

Answer: x = 78 / 341

Step 1: Identify the unknown

Let x be the number we are looking for.

Step 2: Translate the verbal description into a mathematical expression

First, we add 6 to the number x (which gives x + 6). Next, we multiply this result by 13: (x + 6) * 13. Then, we divide the previous result by 6 times x: ((x + 6) * 13) / (6 * x). Finally, we equate the entire result to 59: ((x + 6) * 13) / (6 * x) = 59.

Step 3: Simplify the equation and solve for x

To solve the equation, first, we can multiply both sides by 6x to get rid of the denominator: (x + 6) * 13 = 59 * (6 * x). Now, distribute the 13 and the 6 * 59: 13x + 78 = 354x. Next, subtract 13x from both sides to isolate x terms on one side: 78 = 341x. Finally, to solve for x, divide both sides by 341: x = 78 / 341.

Key Concepts

Translation of Mathematical Phrases

When translating mathematical phrases into algebraic expressions, it's essential to break down the words into mathematical symbols and operations. This often involves recognizing keywords that indicate specific actions.

For example, the phrase "a number is added to six" translates to \( x + 6 \). Here, "a number" is an unknown value, often represented as the variable \( x \). Additionally, "added to" indicates a summation.

Similarly, when we encounter "that result is multiplied by thirteen," it means taking our previous result, \( (x + 6) \), and performing multiplication, resulting in \((x + 6) \cdot 13\). Furthermore, to "divide by six times the number" becomes \(\frac{(x + 6) \cdot 13}{6x}\). Finally, setting the expression equal to 59 transforms the entire scenario into an equation \(\frac{(x + 6) \cdot 13}{6x} = 59\).

Recognizing these keywords and their mathematical counterparts is crucial for building the correct algebraic expressions.

Solving Equations

Once we have translated a verbal phrase into a mathematical equation, the next step is to solve it. The process of solving equations involves finding the value of the unknown variable that makes the equation true.

In our example, after translating the sentence into the equation \(\frac{(x + 6) \cdot 13}{6x} = 59\), we begin solving it. First, multiply both sides by the denominator to eliminate the fraction. This calculation yields \((x + 6) \cdot 13 = 59 \cdot 6x\).

Distribute the 13 on the left side and the 59 on the right side:

• Left side: \(13x + 78\)
• Right side: \(354x\)

By subtracting \(13x\) from both sides, we isolate the variable terms on one side: \(78 = 341x\).

Finally, divide each side by 341 to solve for \(x\). With clear steps, one can effectively solve similar algebraic expressions.

Unknown Variables

The concept of unknown variables often refers to the values in mathematical problems that we need to find. In algebra, this unknown is usually represented by a symbol, commonly \( x\).

Using variables allows us to generalize problems and find solutions through equations. In particular, the unknown \( x \) integrates seamlessly into algebraic expressions and represents numbers that satisfy the built equations.

For instance, in our exercise, "a number is added to six" hints at an unknown, which we denote as \( x \). This variable persists throughout our calculation, featuring in expressions such as \((x + 6) \cdot 13\) and \(6x\).

By understanding and manipulating these variables, we can solve for them by following structured steps, like isolating \( x\) and using operations to determine its numerical value. Recognizing and efficiently handling unknowns is a crucial part of algebra.