Problem 160 - Elementary Algebra

Question

Because of road construction in one city, commuters were advised to plan that their Monday morning commute would take \(150 \%\) of their usual commuting time. Explain what this means.

Step-by-Step Solution

Answer

The commute will take 1.5 times longer than usual.

Step 1 - Understanding 150%

To understand 150%, break it down. 150% means 150 out of 100, or one and a half times as much.

Step 2 - Interpret Usual Commuting Time

Identify the usual commuting time as the base amount. Let’s denote the usual commuting time as 'T'.

Step 3 - Calculate New Commuting Time

To find the new commuting time, multiply the usual commuting time by 1.5. This is because 150% is equal to 1.5 in decimal form.

Step 4 - Express the Calculation

Express the calculation with the formula: \[ \text{New Commuting Time} = 1.5 \times T \]

Step 5 - Conclude the New Time

The new commute will take 1.5 times the usual commuting time due to the construction.

Key Concepts

percentage calculations

Calculating percentages is a crucial skill in algebra. A percentage represents a fraction of 100. When we deal with percentages greater than 100%, it means we have more than the whole. For example, 150% translates to 150 out of 100, or 1.5.

To find a percentage of any number (like the usual commuting time in our problem), follow these steps:

• Convert the percentage to decimal form by dividing by 100 (e.g., 150% becomes 1.5).
• Multiply the base amount by this decimal. In our case, the base is the usual commuting time.

Applying the steps to this problem:

• Usual time is T.
• 150% of T equals 1.5 * T.

So, the new commuting time here is 1.5 times the usual time.

interpretation of percentages

Understanding percentages in real-world contexts is key. In this exercise, the city suggested commuters expect their travel time to be 150% of their usual. This implies an increase. Here, a 100% increase would mean the travel time doubles. But since it's 150%, it’s an additional half of the usual time more.

If usual time is T, here’s the breakdown:

• 100% of T is just T (the same time).
• 150% of T includes 100% (T) plus another 50% (0.5*T).

Combining them, we get 1.5 * T, showing that the journey will take 1.5 times longer due to the construction.

multiplication in algebra

In algebra, multiplication helps scale quantities. Here we used it to scale time. If T represents the usual commuting time, multiplying T by 1.5 gives the new time considering the percentage increase.

Expressed mathematically:

• Identify the base amount (T).
• Convert the percentage into a decimal (150% = 1.5).
• Multiply the base amount by the decimal (1.5 * T).

Using algebraic multiplication in percentage problems lets you efficiently find results. Always remember, understanding the context and setting up your problem correctly are the first steps. Then follow through with the arithmetic!