Question

Solve for $x$ in the following proportions. Carry division two decimal places as necessary. $1:x=5:200$

Short answer

The value of $x$ is 40.

Step-by-step solution

Understand the Proportion

The equation you're dealing with is a proportion: $\frac{1}{x}=\frac{5}{200}$. This states that the ratio of $1$ to $x$ is the same as the ratio of $5$ to $200$.

Set Up the Equation

To solve for $x$, we can cross-multiply the terms of the proportion. This gives us the equation: $1\times 200=5\times x$.

Perform Cross-Multiplication

Calculate $1\times 200$ to simplify the expression on the left side. This gives us:$$200=5x$$

Solve for x

To isolate $x$, divide both sides of the equation by $5$:$$x=\frac{200}{5}$$

Calculate the Division

Perform the division: $\frac{200}{5}=40$. Therefore, $x=40$.

Key concepts

Cross-multiplication

Cross-multiplication is a powerful technique for solving equations involving proportions. When we say two ratios are equivalent, such as $\frac{1}{x}=\frac{5}{200}$, we mean that the two fractions represent the same value. To solve for $x$ here, we can use cross-multiplication. This involves multiplying across the equals sign in a criss-cross manner. This method is based on the cross-products of the fractions being equal:

• The numerator of the first fraction is multiplied by the denominator of the second ($1\times 200$)
• The denominator of the first fraction is multiplied by the numerator of the second ($x\times 5$)

By setting these two products equal, we get the equation $200=5x$. Cross-multiplying simplifies your equation-solving, making it a go-to tool for handling proportion problems effectively. Ensure you perform this operation accurately to maintain the balance of the equation.

Solving equations

Solving equations often involves isolating the variable you're looking for, which is $x$ in this case. After cross-multiplying, our equation becomes $200=5x$. This is a straightforward linear equation. To solve for $x$, focus on isolating $x$ on one side of the equation. We want to get $x$ by itself to discover its value.

• Our main goal here is to remove whatever is attached to $x$ using inverse operations.
• Since $x$ is multiplied by 5, we will use division to cancel this multiplication, effectively isolating $x$ on its side of the equation.

The resulting equation once solved will reveal the value of $x$. Solving equations via such methods is crucial in algebra since it forms the basis for understanding more complex mathematical concepts.

Division steps

Once we have isolated $x$, which resulted in the equation $x=\frac{200}{5}$, the next step is to perform the division. Here is how you can dissect the division step:

• Start by dividing the numerator by the denominator: $200÷5$.
• Think about how many times 5 goes into 200. This is a simple division problem where 5 fits into 20 four times.
• Therefore, $5\times 40=200$, confirming that $x=40$.

Dividing gets us down to a precise answer for $x$ and is an essential step when solving proportion problems. Practice these steps to feel more at ease with arithmetic operations, especially when precision matters and answers need to be rounded to two decimal points if necessary.