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Magazine Chapter 7: Problem 66
The quotient of a number and \(5,\) minus \(1,\) equals \(\frac{7}{5}\). Find the number.
Short Answer
Expert verified
The number is 12.
Step by step solution
01
Set Up the Equation
The problem states that the quotient of a number and 5, minus 1, equals \( \frac{7}{5} \). Let the unknown number be \( x \). We can create the equation: \( \frac{x}{5} - 1 = \frac{7}{5} \).
02
Isolate the Fraction
To solve for \( x \), we first need to get rid of the \(-1\) by adding 1 to both sides of the equation: \( \frac{x}{5} - 1 + 1 = \frac{7}{5} + 1 \). This simplifies to \( \frac{x}{5} = \frac{7}{5} + \frac{5}{5} \).
03
Combine the Fractions
Add the fractions on the right side: \( \frac{7}{5} + \frac{5}{5} = \frac{12}{5} \). Now the equation is \( \frac{x}{5} = \frac{12}{5} \).
04
Solve for the Number
To find \( x \), multiply both sides by 5, cancelling out the denominator: \( x = \frac{12}{5} \times 5 \). Simplifying, we get \( x = 12 \).
05
Verify the Solution
Plug \( x = 12 \) back into the original equation to verify: \( \frac{12}{5} - 1 = \frac{7}{5} \). Simplifying, \( \frac{12}{5} - \frac{5}{5} = \frac{7}{5} \), which confirms our solution is correct.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Solving Equations
Solving equations is like finding the missing piece of a puzzle. In this exercise, we aim to find the unknown number, represented by the variable \(x\). An equation is a mathematical statement that shows that two expressions are equal, using an equal sign.
For instance, in the equation \(\frac{x}{5} - 1 = \frac{7}{5}\), the left side represents the original problem condition and the right side is the given result.
The process involves applying operations step-by-step to both sides of the equation to isolate and solve for the unknown variable.
For instance, in the equation \(\frac{x}{5} - 1 = \frac{7}{5}\), the left side represents the original problem condition and the right side is the given result.
The process involves applying operations step-by-step to both sides of the equation to isolate and solve for the unknown variable.
- First, we simplify the equation using basic operations such as addition or subtraction. This balances both sides.
- Next, we use multiplication or division to isolate the variable.
- Finally, we verify the solution by plugging it back into the original equation, ensuring correctness.
Fractions
Fractions are a way to represent parts of a whole and are often encountered in algebra problems. Understanding fractions is key to solving equations involving them.
In our example, we encounter fractions several times: \(\frac{7}{5}\), \(\frac{5}{5}\), and \(\frac{12}{5}\). These fractions illustrate parts of division results or how expressions are used together to form whole values.
Here鈥檚 what you'll frequently do when handling fractions:
In our example, we encounter fractions several times: \(\frac{7}{5}\), \(\frac{5}{5}\), and \(\frac{12}{5}\). These fractions illustrate parts of division results or how expressions are used together to form whole values.
Here鈥檚 what you'll frequently do when handling fractions:
- Identify the common denominators, which allow for the addition or subtraction of fractions on the same side of the equation.
- Combine fractions by summing up their numerators and keeping the denominator constant when they have an identical denominator.
- Simplify fractions into whole numbers or simpler fractions when possible.
Variable Isolation
Variable isolation is about getting the variable on one side of the equation all by itself. It is a crucial skill in solving algebraic equations, and here's how it works in this example:
Initially, we had the equation \(\frac{x}{5} - 1 = \frac{7}{5}\). Our aim was to make \(x\) stand alone.
The steps for isolating a variable usually include:
Initially, we had the equation \(\frac{x}{5} - 1 = \frac{7}{5}\). Our aim was to make \(x\) stand alone.
The steps for isolating a variable usually include:
- Remove any constants or numbers attached to the variable using inverse operations. For example, we added 1 to both sides to cancel out the \(-1\) next to \(\frac{x}{5}\).
- Eliminate fractions by multiplying by denominators, as we multiplied by 5 to resolve \(\frac{x}{5}\) into just \(x\).
- Maintain balance by doing the same operation to both sides of the equation.
