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Chapter 6: Problem 116

Evaluate \(\frac{250 x}{100-x}\) for \(x=60\)

Short Answer

Expert verified
The solution of \(\frac{250x}{100-x}\) for \(x=60\) is \(375\).

Step by step solution

01

Substitution

Begin the problem by substituting the given value \(x = 60\) into the expression \(\frac{250x}{100-x}\). The equation should now appear as \(\frac{250*60}{100-60}\).
02

Simplification

Simplify the equation \(\frac{250*60}{100-60}\) by first performing the multiplication and the subtraction in the numerator and denominator respectively. This leads to \(\frac{15000}{40}\).
03

Division

Finally, perform the division \(15000 梅 40\).

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Key Concepts

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

Understanding Substitution
Substitution is a technique used in algebra to replace a variable with a specific value. This is often applied in evaluating algebraic expressions. When given a value for a variable, you simply replace the variable with this number.
For instance, if the expression is \( \frac{250x}{100-x} \) and you're asked to evaluate it for \( x = 60 \), substitution means you replace \( x \) with 60.
  • This transformation helps turn an abstract expression into a concrete number.
  • Once substituted, the expression takes a simpler form that can be worked with numerically.
Applying substitution here gives us \( \frac{250 \times 60}{100-60} \).
The key is to always replace correctly and verify the expression before further steps.
Simplification Strategies
Simplification in algebra involves making an expression into its simplest form. After substitution, we often face complex expressions that require simplification before solving.
  • This process may involve arithmetic operations such as addition, subtraction, multiplication, and division.
  • In our problem, after substituting, we need to multiply the numbers and subtract the terms as per arithmetic rules.
For the expression \( \frac{250 \times 60}{100-60} \), start by solving the arithmetic:
  • Numerator: \( 250 \times 60 = 15000 \)
  • Denominator: \( 100 - 60 = 40 \)
This renders the expression as \( \frac{15000}{40} \). Ensuring all operations are precise supports finding the correct outcome.
Performing Division
Division is an arithmetic operation where you determine how many times one number is contained within another. This is often the final step in solving algebraic expressions after simplification.
For the given problem, once we simplify to \( \frac{15000}{40} \), the next logical step is to perform division:
  • Divide 15000 by 40.
  • When dividing, one useful method is long division, or directly calculating using a calculator.
Through division, we find that 15000 divided by 40 equals 375.
Thus, performing division gives you the final numerical result of an algebraic expression after substitution and simplification.

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