Problem 53 - Elementary Algebra

Question

For the following problems, translate the following phrases or sentences into mathematical expressions or equations. An unknown quantity is decreased by eleven. This result is then divided by fifteen. Now, one is subtracted from this result and five is obtained.

Step-by-Step Solution

Answer

Answer: ((x - 11) / 15) - 1 = 5

Step 1: Identify the unknown quantity

Let x represent the unknown quantity.

Step 2: Decrease the unknown quantity by eleven

The problem states that an unknown quantity is decreased by eleven. We can write this as a mathematical expression: x - 11

Step 3: Divide the result by fifteen

The problem states that the result from step 2 is divided by fifteen. Using the expression from step 2, we can write this as: (x - 11) / 15

Step 4: Subtract one from the result

The problem states that one is subtracted from the result of step 3. Using the expression from step 3, we can write this as: ((x - 11) / 15) - 1

Step 5: Set the final result equal to five

Finally, the problem states that the result of step 4 is equal to five. Using the expression from step 4, we can write this as an equation: ((x - 11) / 15) - 1 = 5 Now, we have translated the word problem into a mathematical equation.

Key Concepts

Algebraic Expressions

Algebraic expressions are a fundamental element in understanding and solving mathematical problems. They are combinations of numbers, variables, and arithmetic operations (such as addition, subtraction, multiplication, and division) without an equal sign. For instance, when a problem describes 'an unknown quantity decreased by eleven,' we translate it into an algebraic expression as \( x - 11 \).

In the educational exercise, the phrase 'This result is then divided by fifteen' further operates on the initial expression, resulting in \( \frac{x - 11}{15} \). The beauty of algebraic expressions lies in their ability to represent complex ideas simply and symbolically. As students work through exercises, they learn to manipulate these expressions by applying mathematical operations to uncover the value of the unknown quantities they represent.

Unknown Quantity

An 'unknown quantity' in mathematics is often symbolized by a variable, commonly denoted as \( x \) or another letter, and represents a number that has yet to be determined. Word problems frequently introduce an unknown quantity that we are asked to solve for. The ability to identify and appropriately express this unknown is crucial in the translation process from a word problem to an algebraic equation.

In the educational scenario, we start by denoting the unknown quantity with \( x \). As students progress in algebra, identifying and representing unknown quantities becomes almost second nature, allowing for the straightforward construction and deconstruction of more complex algebraic statements and equations.

Mathematical Equations

Mathematical equations are statements that assert the equality of two expressions. They are comprised of two algebraic expressions set equal to each other and are often used to determine the value(s) of unknown quantities. In problems that require translating words to equations, the final step involves equating an algebraic expression to a specific value, as guided by the textual description.

In our provided solution, translating the final part of the problem '...and five is obtained' leads us to construct the equation \( \left(\frac{x - 11}{15}\right) - 1 = 5 \). Solving this equation will reveal the value of \( x \), giving concrete meaning to the initially abstract exercise. Equations allow us to visualize and solve real-life problems numerically, which is a powerful aspect of learning mathematics.