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Chapter 2: Problem 37

Solve each equation. $$ \frac{k}{4}=\frac{175}{20} $$

Short Answer

Expert verified
k = 35

Step by step solution

01

Simplify the right side

First, simplify the fraction on the right side of the equation \ \[ \frac{175}{20} \ \]. \ This can be done by dividing both the numerator and the denominator by their greatest common divisor (GCD). In this case, the GCD of 175 and 20 is 5. \ Therefore: \ \[ \frac{175}{20} = \frac{175 \div 5}{20 \div 5} = \frac{35}{4} \].
02

Set the equation

After simplifying, set the equation with the simplified fraction: \ \[ \frac{k}{4} = \frac{35}{4} \].
03

Cross-multiply

Since the fractions are equal, we can cross-multiply to solve for \(k\). This means multiplying the numerator of one fraction by the denominator of the other: \ \[ 4 \cdot \frac{35}{4} = 35 \]. \ Therefore, \ \[ k = 35 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fraction Simplification
Simplifying fractions is an important skill in algebra.
By simplifying, we reduce the fraction to its simplest form.
This makes it easier to work with and understand.

To simplify a fraction, find the greatest number that evenly divides both the numerator and the denominator. This number is known as the Greatest Common Divisor (GCD).
  • For the fraction \(\frac{175}{20}\), first identify the GCD of 175 and 20, which is 5.
  • Next, divide both the numerator and the denominator by the GCD.
  • So, we get \(\frac{175 \div 5}{20 \div 5} = \frac{35}{4}\).
  • Now, \(\frac{35}{4}\) is the simplified form of \(\frac{175}{20}\).
Simplified fractions make further calculations easier and more manageable.
Cross-Multiplication
Cross-multiplication is a handy method to solve equations involving fractions.
When two fractions are set equal to each other,
you can cross-multiply to find the value of the unknown variable.
  • Consider the equation \(\frac{k}{4} = \frac{35}{4}\).
  • Cross-multiplying involves multiplying the numerator of one fraction by the denominator of the other and vice versa.
  • In this case, multiplying \(k\) by 4 and 4 by 35.
  • This gives us \(4 \cdot \frac{35}{4} = 35\).
Thus, we find that \(k = 35\).
Cross-multiplication is particularly useful because it clears the fractions, making the equation easier to solve.
Greatest Common Divisor
The Greatest Common Divisor (GCD) is the largest number that can divide both the numerator and denominator of a fraction evenly.
Using the GCD simplifies the fraction to its smallest, simplest form.
  • For instance, consider the numbers 175 and 20.
  • The GCD of 175 and 20 is 5.
  • This is because 5 is the largest number that divides both 175 and 20 without leaving a remainder.
  • By dividing both the numerator and the denominator of \(\frac{175}{20}\) by their GCD, we get \(\frac{35}{4}\).
Understanding how to find the GCD is crucial for fraction simplification.
It helps make complex algebraic equations much simpler to solve.

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Most popular questions from this chapter

Solve each inequality. Graph the solution set, and write it using interval notation. \(-4<-2 x<12\)

U.S. five-cent coins are made from a combination of nickel and copper. For every \(1 \mathrm{lb}\) of nickel, 3 lb of copper are used. How many pounds of each metal would be needed to make 560 lb of five-cent coins? (Data from The United States Mint.)

When 6 is added to \(\frac{3}{4}\) of a number, the result is 4 less than the number. Find the number.

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