Question

Solve for \(x\) in the following proportions. Carry division two decimal places as necessary. \(\frac{1}{x}=\frac{10}{6}\)

Short answer

The solution for \(x\) is 0.60.

Step-by-step solution

Understand the Equation

We are given the proportion \(\frac{1}{x} = \frac{10}{6}\). A proportion indicates that two ratios are equal.

Cross-Multiply to Remove the Fractions

To solve for \(x\), we can cross-multiply. This means multiplying the numerator of each fraction by the denominator of the other:\[1 \times 6 = 10 \times x\] which simplifies to:\[6 = 10x\].

Isolate the Variable x

To solve for \(x\), divide both sides of the equation by 10:\[x = \frac{6}{10}\].

Simplify the Fraction

Divide 6 by 10 to get a decimal, carrying division to two decimal places as necessary:\[x = 0.60\]Since 6 divided by 10 gives exactly 0.6, the value at two decimal places is 0.60.

Key concepts

Understanding Proportions

When we talk about proportions, we discuss the idea that two ratios or fractions are equal. In math, a proportion will look like two fractions set equal to each other. For example, in our exercise, \( \frac{1}{x} = \frac{10}{6} \), the expression means that the ratio of 1 to \(x\) is the same as the ratio of 10 to 6. This means whatever operation we perform on one side of the equation should be similarly performed on the other to maintain the equality.

Proportions are useful because they allow us to solve for an unknown variable in one of the ratios. By understanding the relationships between the quantities involved, we can determine the missing value.

The Power of Cross-Multiplication

Cross-multiplication is a powerful algebraic technique used to solve proportions. It simplifies dealing with fractions by eliminating them. When you cross-multiply, you multiply the numerator of one fraction by the denominator of the other, essentially 'crossing' them.

In the equation \( \frac{1}{x} = \frac{10}{6} \):

• Multiply \(1\) (numerator of the first fraction) by \(6\) (denominator of the second fraction),
• Multiply \(10\) (numerator of the second fraction) by \(x\) (denominator of the first fraction).

This gives us:\[1 \times 6 = 10 \times x\]which simplifies to:\[6 = 10x\]

Cross-multiplication effectively removes the fraction and converts the equation into a simpler format that is easier to solve for the unknown variable.

Using Algebra to Solve Equations

Algebra is a fundamental part of solving equations like the one in the original problem. Once we apply cross-multiplication, we are left with an equation without fractions \(6 = 10x\). The next step is to solve for the variable, in this case, \(x\).

To isolate \(x\), you will need to perform an inverse operation. Here, \(x\) is being multiplied by \(10\), so to get \(x\) alone, divide both sides of the equation by \(10\):\[x = \frac{6}{10}\]
This operation keeps the equation balanced, meaning whatever you do to one side of the equation, you should do to the other.

Performing Decimal Division

In this problem, once we isolate \(x\), we need to convert the fraction \(\frac{6}{10}\) into a decimal by division. To do so, simply divide the numerator by the denominator.

Here, when you divide 6 by 10, you get \(x = 0.6\).

Since the division results in a terminating decimal, you can express this as \(0.60\) when rounded to two decimal places, even though no rounding is needed in this specific example as it equals \(0.6\) exactly. Rounding to two decimal places is crucial when required, as it ensures precision in calculations, especially in contexts where accuracy is vital.