Problem 34 - Basic College Mathematics

Question

Solve. \(\frac{3}{2} \cdot t=90\)

Step-by-Step Solution

Answer

The solution is \(t = 60\).

Step 1: Write the equation

Start by writing down the given equation: \(\frac{3}{2} \cdot t = 90\)

Step 2: Isolate the variable

To solve for \(t\), you need to isolate it on one side of the equation. Divide both sides by \( \frac{3}{2} \): \[\frac{\frac{3}{2} \cdot t}{\frac{3}{2}} = \frac{90}{\frac{3}{2}}\]

Step 3: Simplify the right-hand side

To divide by a fraction, multiply by its reciprocal: \[t = 90 \cdot \frac{2}{3}\]

Step 4: Perform the multiplication

Calculate the multiplication on the right-hand side: \[t = 90 \cdot \frac{2}{3} = 60\]

Step 5: Write the final answer

Now that the calculations are done, write the final value of \(t\): \[t = 60\]

Key Concepts

fractions

When working with fractions, it's essential to understand their basic structure. A fraction has two parts: the numerator (top number) and the denominator (bottom number). Fractions represent a part of a whole.
In solving equations, fractions can sometimes seem tricky, but they follow simple rules. For instance, multiplying by a fraction means you multiply the numerator and denominator separately.

To divide by a fraction, you do not actually divide; instead, you multiply by its reciprocal. The reciprocal of a fraction is obtained by swapping its numerator and denominator. For example, the reciprocal of \(\frac{3}{2}\) is \(\frac{2}{3}\). Using the reciprocal helps in isolating variables in equations involving fractions.

isolating variables

Isolating the variable is a critical step in solving equations. It means rearranging the equation so that the variable you're solving for stands alone on one side.

In the equation \(\frac{3}{2} \cdot t = 90\), to isolate \(t\), we need to remove the fraction next to it. We do this by performing the inverse operation. Since \(\frac{3}{2}\) is multiplying \(t\), we divide both sides by \(\frac{3}{2}\): \[ \frac{\frac{3}{2} \cdot t}{\frac{3}{2}} = \frac{90}{\frac{3}{2}} \].
Dividing by a fraction is equivalent to multiplying by its reciprocal. This simplifies to: \[ t = 90 \cdot \frac{2}{3} \]
Here, \(t\) is isolated, we've moved towards finding its value.

multiplication

Multiplication is one of the basic arithmetic operations and involves combining equal groups. When multiplying fractions, you multiply the numerators together and the denominators together.

In the equation given, after isolating the variable \(t\), we had: \[ t = 90 \cdot \frac{2}{3} \]

• First, you multiply the numerators: \(90 \times 2 = 180\).
• Then, you divide the product by the denominator: \(\frac{180}{3} = 60\).

The final solution of our equation is \(t = 60\). Simplifying correctly is essential to arriving at the right answer.