Question

Set up the following problems as a proportion and solve. Include labels in the set up and on the answer. If 1 kilogram (kg) equals \(2.2 \mathrm{lb}\), how may \(\mathrm{kg}\) are in \(61.6 \mathrm{lb}\) ?

Short answer

61.6 lb is 28 kg.

Step-by-step solution

Understand the problem

We need to find out how many kilograms are equivalent to 61.6 pounds using the given conversion factor of 1 kg = 2.2 lb.

Set up the proportion

We can set up a proportion where the ratio of kg to lb is the same on both sides of the equation. Let the unknown number of kg be represented by \( x \): \[ \frac{1 \, \text{kg}}{2.2 \, \text{lb}} = \frac{x \, \text{kg}}{61.6 \, \text{lb}} \]

Cross-multiply to solve for x

To solve for \( x \), cross-multiply: \[ 1 \, \text{kg} \times 61.6 \, \text{lb} = 2.2 \, \text{lb} \times x \, \text{kg} \] This simplifies to: \[ 61.6 = 2.2x \]

Calculate x

Solve the equation for \( x \) by dividing both sides by 2.2: \[ x = \frac{61.6}{2.2} \] Computing the division gives: \[ x = 28 \] So, \( 61.6 \, \text{lb} \) is equivalent to \( 28 \, \text{kg} \).

Key concepts

Proportions

Proportions are a powerful way to compare two ratios or comparisons between numbers. When dealing with measurements, a proportion is a statement that two ratios are equal. For example, if we have a conversion rate, like knowing that 1 kilogram is equal to 2.2 pounds, we can express this as a proportion. This allows us to find an unknown value if we know one part of the proportion. By setting up proportions, we create an equation that can be solved by keeping the two parts of the equation balanced. Keeping labels in your proportions can help you keep track of units and ensure you set up your equation correctly. In our specific problem, the proportion allows us to convert pounds to kilograms seamlessly.

Conversion Factors

A conversion factor is a number used to change one set of units to another, by multiplying or dividing. This is especially useful when converting measurements from one system to another, like pounds to kilograms in our scenario. The conversion factor for our exercise is given as 1 kg = 2.2 lb. This means whenever we have pounds and we want kilograms, we can divide by 2.2 to switch the units.

• Conversion factors need to be accurate to produce precise results.
• They help simplify calculations involving different measurement systems.

In the example, knowing the conversion factor allows you to set up the correct equation and solve for the unknown weight in kilograms.

Solving Equations

Solving equations is the process of finding a value for a variable that makes the equation true. In proportion problems, this typically involves finding the unknown value that balances both sides of the equation. After setting up an equation from the proportion, we can use simple algebra to solve for the variable.
To solve the equation, we:

• Cross-multiply to eliminate the fractions. This involves multiplying across the equals sign in a diagonal pattern.
• Isolate the variable by performing inverse operations, like division or multiplication.

For our problem, after setting the proportion, we cross-multiply and divide to isolate and solve for the kilograms, resulting in 28 kg for 61.6 lb. Using these steps makes solving proportion problems straightforward and manageable.