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Chapter 6: Problem 39

Solve. Round to the nearest hundredth. $$\frac{7}{10}=\frac{3+n}{2}$$

Short Answer

Expert verified
-1.6

Step by step solution

01

Simplify the equation

The first step is to simplify the equation. This can be done by multiplying both sides of the equation by 2 to get rid of the fraction on the right side of the equation. This will transform the equation to \(1.4 = 3 + n\).
02

Isolate the variable

Next, to isolate 'n', subtract 3 from both sides of the equation to get \( n = 1.4 - 3\). This simplifies to \(n = -1.6\).
03

Round the answer to the nearest hundredth

The final answer, -1.6, is already rounded to the nearest hundredth.

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Key Concepts

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

Equation Solving
Equation solving involves finding the value of an unknown that makes a mathematical statement true. In this exercise, we started with a fractional equation: \( \frac{7}{10} = \frac{3+n}{2} \). To solve it, we took the following steps:
  • Clear the fraction: Fractions can complicate equations, so it helps to eliminate them. We do this by multiplying each side of the equation by the denominator of the fraction we're trying to clear. In this case, multiplying both sides by 2 helps to clear the fraction on the right, resulting in \(1.4 = 3 + n\).

  • Isolate the variable: Our goal is to find the value of \(n\). To do this, we need to get \(n\) by itself on one side of the equation. We subtract 3 from both sides, leading to \(n = 1.4 - 3\).

  • Solution: The result of this subtraction is \(n = -1.6\).
Breaking down each operation ensures the equation stays balanced and helps in gradually leading to \(n\)'s solution.
Fraction Simplification
Fraction simplification plays a crucial role in handling complex equations. A fraction is a way to represent a part of a whole or a division of numbers. In this problem, the fraction \(\frac{3+n}{2}\) was simplified by eliminating the denominator:
  • Multiply both sides: To simplify fractions in equations, multiply every term by the least common multiple of the denominators involved. Here, multiplying by 2 simplified the equation.

  • Resulting equation: The multiplication helps to transform our equation into \(1.4 = 3 + n\), removing the fraction.
This simplification makes the equation easier to solve by eliminating the fractional component.
Rounding Numbers
Rounding numbers is a common practice especially when dealing with decimals to simplify mathematical expressions and results. In this exercise, we encountered a rounded value after solving for \(n\):
  • Understanding rounding: Rounding is the process of approximating a number to a specified degree of accuracy. Commonly, we round numbers to the nearest whole number, tenth, or hundredth.

  • Rounding to the nearest hundredth: If the number we look at is in the thousandths place, and it's 5 or greater, the hundredths digit increases by 1. If it's less than 5, the hundredths digit stays the same. Here, \(-1.6\) is already rounded to the nearest hundredth since it doesn't have more decimal places.
Rounding ensures results are meaningful and manageable within the context, making solutions easier to interpret and communicate.

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