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Chapter 6: Problem 78

Solve. $$ 3=0.16 \times b $$

Short Answer

Expert verified
The value of b is 18.75.

Step by step solution

01

Understand the Equation

The given equation is \[ 3 = 0.16 \times b \]. The goal is to solve for the variable \( b \).
02

Isolate the Variable \( b \)

To isolate \( b \), divide both sides of the equation by 0.16. This will give: \[ b = \frac{3}{0.16} \]
03

Perform the Division

Now, perform the division to find the value of \( b \). Calculate \( \frac{3}{0.16} = 18.75 \).
04

Verify Your Solution

Check the calculation by multiplying 0.16 by 18.75 to see if it equals 3. Indeed, \( 0.16 \times 18.75 = 3 \). This confirms our solution is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating Variables
When solving linear equations, the primary goal is to isolate the variable you are solving for. In our example, the variable is \(b\). To isolate \(b\), you need to remove any coefficients or constants attached to it by performing inverse operations on both sides of the equation.

In the equation \(3 = 0.16 \times b\), \(b\) is being multiplied by 0.16. The inverse operation of multiplication is division. Therefore, to isolate \(b\), you will divide both sides of the equation by 0.16:

\ \[ b = \frac{3}{0.16} \] \
This step ensures that \(b\) stands alone on one side of the equation, making it easier to find its value.
Decimal Division
Performing division with decimals can sometimes be tricky, but it follows the same principles as dividing whole numbers. In our solved equation, we have:

\ \[ b = \frac{3}{0.16} \] \
To carry out this division accurately, you can use a calculator, or you can convert the decimal into a fraction and then perform the division. Here's a simple calculator approach:
  • You type in 3 divided by 0.16.
  • The result is 18.75.

Thus, \( b = 18.75 \). This method helps you avoid errors and ensures precision in your calculations.
Verifying Solutions
After finding the solution, it's crucial to verify its accuracy. This step ensures that no mistakes were made during the calculation process. For our equation, we found\( b = 18.75 \). To verify, substitute 18.75 back into the original equation:

\ \[ 3 = 0.16 \times 18.75 \] \
Now multiply 0.16 by 18.75:
  • Calculate 0.16 times 18.75.
  • The result is exactly 3.

Since the left side of the equation equals the right side when \(b = 18.75\), our solution is confirmed to be correct. Verifying solutions helps build confidence and ensures that the problem-solving process was executed correctly.

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Most popular questions from this chapter

The effective yield is the yearly rate of simple interest that corresponds to a rate for which interest is compounded two or more times a year. For example, if \(P\) is invested at \(12 \%,\) compounded quarterly, we multiply \(P\) by \((1+0.12 / 4)^{4},\) or \(1.03^{4}\). Since \(1.03^{4} \approx 1.126\), the \(12 \%\) compounded quarterly corresponds to an effective yield of approximately \(12.6 \% .\) In Exercises 47 and \(48,\) find the effective yield for the indicated account. The account pays \(10 \%\) compounded daily.

Solve. Shear Delights Hair Salon borrows \(\$ 3600\) at \(4 \%\) for 30 days. Find (a) the amount of interest due and (b) the total amount that must be paid after 30 days.

Translate to an equation and solve. What percent of 60 is \(15 ?\)

Find decimal notation. $$ 0.12 \% $$

Translate to a proportion. Do not solve. $$ 66 \% \text { of } 74 \text { is what? } $$
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