Problem 26 - Algebra 1: Concepts and Skills

Question

Solve the percent problem. 30 grams is 20% of what weight?

Step-by-Step Solution

Answer

The total weight is 150 grams.

Step 1: Set up the equation

From the problem, we can write our equation as \(0.20 = \frac{30}{x}\). This represents 20% (0.20 in decimal format) of the total weight (x) equals 30 grams (which is the given part).

Step 2: Solve for x

By multiplying both sides by x, we eliminate x from the denominator on the right side of our equation. This results in \(0.20x = 30\). Dividing by 0.20 to isolate x on one side, we perform \(x = \frac{30}{0.20}\)

Step 3: Calculate the value of x

By executing the calculation in step 2, we determine that \(x = 150\) grams. This is the total weight where 30 grams represents 20%.

Key Concepts

Equations

Equations are fundamental tools in mathematics for finding unknown values using known information. Think of them as mathematical sentences.
They express a relationship between different values. In the given percentage problem, the equation is set up as follows:

• We know that 30 grams is 20% of some total weight, which we’ll call \( x \).
• The equation starts with writing down this relationship as \( 0.20 = \frac{30}{x} \).

This equation states that 20% of the total weight \( x \) is equivalent to 30 grams.
When solving an equation like this, our goal is to isolate the unknown variable (\( x \)) on one side. This process typically involves operations such as multiplication, division, addition, or subtraction. Here, to eliminate \( x \) from the denominator, we multiply both sides by \( x \), transforming our equation to \( 0.20x = 30 \). Through further manipulation, we find the value of \( x \).

Percentages

Percentages are all around us, used to express proportions, comparisons, and changes. They are essentially parts per hundred. In this problem, understanding percentages is key to setting up the initial equation.
A percentage represents a decimal fraction. For example, 20% can be converted to a decimal by dividing by 100, resulting in 0.20. This helps when setting up equations since decimals allow for algebraic manipulation.

• To find what the total weight (\( x \)) is, given 30 grams is 20% of it, we express the relation as \( 0.20 = \frac{30}{x} \).
• By converting percentages into decimals, we simplify calculations and facilitate solving equations algebraically.

This conversion is a fundamental first step when tackling percentage problems, as it paves the path for further algebraic steps.

Algebra

Algebra extends our basic arithmetic skills into new territory, allowing us to handle variables and unknowns. In this problem, algebra helps us solve for the total weight \( x \).

• We start from the equation \( 0.20 = \frac{30}{x} \), derived from our percentage setup.
• Algebra involves manipulating this equation to find \( x \). We adjust it, multiplying both sides by \( x \) to clear the denominator, forming \( 0.20x = 30 \).
• The next step is isolating \( x \). By dividing both sides by 0.20, we solve for \( x \), yielding \( x = \frac{30}{0.20} \).
• Finally, executing this operation gives us \( x = 150 \), the total weight where 30 grams is 20%.

Algebra simplifies finding unknown values in percent problems, a technique that is versatile and applicable beyond percentage calculations.