Cambio porcentual en pares de números

Problem

Find the percent change for each number pair. Label each change as an increase or a decrease:

$$300 \to 258,\quad 500 \to 435,\quad 600 \to 642,\quad 600 \to 618,\quad 675 \to 729,\quad 935 \to 748$$

Step 1: Identify Original and New Values

In each pair, the first number is the original value and the second number is the new value.

If the new value is larger than the original value, the percent change is an increase. If the new value is smaller than the original value, the percent change is a decrease.

Step 2: Find the Amount of Change

Subtract the smaller number from the larger number to find the amount of change.

For the pair $300 \to 258$:

$$300 - 258 = 42$$

So the change amount is $42$.

Step 3: Make a Change Fraction

Compare the change amount to the original value by writing:

$$\frac{\text{change amount}}{\text{original value}}$$

For $300 \to 258$, the fraction is:

$$\frac{42}{300}$$

Step 4: Convert the Fraction to a Percent

Divide the numerator by the denominator to get a decimal, then multiply by $100$ to get a percent.

$$\frac{42}{300} = 0.14$$ $$0.14 \times 100 = 14\%$$

So the percent change from $300$ to $258$ is $14\%$.

Step 5: Attach Increase or Decrease

Compare the new value to the original value.

Since $258$ is less than $300$, the change from $300$ to $258$ is a $14\%$ decrease.

Step 6: Repeat for Each Pair

Using the same process, place the change amount over the original value and convert to a percent.

For $500 \to 435$:

$$500 - 435 = 65$$ $$\frac{65}{500} = 0.13 = 13\%$$

Since $435 < 500$, this is a $13\%$ decrease.

For $600 \to 642$:

$$642 - 600 = 42$$ $$\frac{42}{600} = 0.07 = 7\%$$

Since $642 > 600$, this is a $7\%$ increase.

For $600 \to 618$:

$$618 - 600 = 18$$ $$\frac{18}{600} = 0.03 = 3\%$$

Since $618 > 600$, this is a $3\%$ increase.

For $675 \to 729$:

$$729 - 675 = 54$$ $$\frac{54}{675} = 0.08 = 8\%$$

Since $729 > 675$, this is an $8\%$ increase.

For $935 \to 748$:

$$935 - 748 = 187$$ $$\frac{187}{935} = 0.20 = 20\%$$

Since $748 < 935$, this is a $20\%$ decrease.

Step 7: List the Final Answers

The percent changes are:

$$300 \to 258:\ 14\% \text{ decrease}$$ $$500 \to 435:\ 13\% \text{ decrease}$$ $$600 \to 642:\ 7\% \text{ increase}$$ $$600 \to 618:\ 3\% \text{ increase}$$ $$675 \to 729:\ 8\% \text{ increase}$$ $$935 \to 748:\ 20\% \text{ decrease}$$

Each answer comes from writing the change amount over the original value and converting that fraction to a percent.