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Chapter 6: Problem 18

For Exercises \(1-20,\) estimate. Then find the actual quotient. $$30.13 \div 0.99$$

Short Answer

Expert verified
Answer: The approximate quotient of 30.13 divided by 0.99 is 30.35.

Step by step solution

01

Estimation of quotient

To estimate the quotient, we can round the numbers to their nearest whole value and then perform the division. We can round \(30.13\) to \(30\) and \(0.99\) to \(1\). Now, the estimated division becomes: $$30 \div 1 = 30$$ So, our estimate for the quotient is 30.
02

Actual division of the given numbers

Now, we will find the actual quotient by dividing \(30.13\) by \(0.99\). To make the divisor a whole number, we can multiply both the dividend and the divisor by 100: $$ \frac{30.13}{0.99}\times \frac{100}{100} = \frac{3013}{99} $$ Now, we can perform the long division to find the actual quotient: $$ \require{enclose} \begin{array}{c|cc cc cc} \multicolumn{2}{r}{3} & . &0 & 3 & 5 \\ \cline{2-6} 99 & \multicolumn{1}{|c}{3} &0& 1 & 3 & \\ \cline{2-3} \multicolumn{2}{r}{1} &1& \\ \cline{3-4} \cline{3-4} \multicolumn{3}{r}{9} & 3 & \\ \cline{4-5} \multicolumn{4}{r}{8} & 7 & \\ \cline{5-6} \multicolumn{5}{r}{}& 5 \\ \end{array} $$
03

Write the final answer

So, the actual quotient of the given problem \(30.13 \div 0.99\) is approximately equal to \(30.35\).

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Key Concepts

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

Estimation
Estimation is a useful technique in mathematics that helps simplify complex calculations by rounding numbers. In the exercise above, we estimate the quotient of dividing two numbers by rounding them to their nearest whole values. For example, we round \(30.13\) to \(30\) and \(0.99\) to \(1\). This gives us an easy calculation:
  • \(30 \div 1 = 30\)
This quick estimation can give us a basic idea of what the actual result might be. It helps in checking our calculations and ensuring that our final answer is reasonable. However, it is important to note that estimation is not the precise answer; it is merely an approximation of the actual quotient.
Long Division
Long division is a method used to divide larger numbers. It breaks down the division process into simpler steps, allowing us to find the actual quotient. Let's look at how long division works with the example of \(30.13 \div 0.99\).
First, we make the divisor a whole number by multiplying both the dividend and the divisor by 100, converting the problem to \(\frac{3013}{99}\).
This is done because dividing by whole numbers is generally easier. We then proceed with a step-by-step division to find our actual quotient:
  • Divide the first portion of the dividend by the divisor to get the first digit of the quotient.
  • Subtract the result from the dividend, bring down the next digit, and repeat.
This method is systematic, reducing errors and providing an accurate result.
Quotient
The quotient is the result obtained from dividing one number by another. In the exercise, after performing the long division, the actual quotient of \(30.13 \div 0.99\) turns out to be approximately \(30.35\).
The process of finding a quotient involves:
  • Estimation to preview the result.
  • Carrying out the necessary calculation (like long division) to find the exact number.
Understanding the quotient is important, as it represents the value of how many times the divisor fits into the dividend. This concept is foundational in many areas of mathematics and is often a basic skill that students should master for more advanced topics.

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Most popular questions from this chapter

Solve. The closing price of a stock one day was 27.90 dollar. The next day the closing price of the same stock was 30.45 dollar. Let \(n\) represent the amount the stock's closing price changed. Write an equation that can be used to find the amount the closing price changed; then solve the equation.

Use \(<\) or \(>\) to make a true statement. $$.2 .891 ? 2.8909$$

Evaluate each expression using the given values. $$\frac{1}{2} m v^{2} ; m=84.6, v=0.2$$

Add or subtract. $$\left(3.1 a^{3}-a^{2}+7.5 a-0.01\right)+\left(6.91 a^{3}-3.91 a^{2}+6\right)$$

Solve. Use the following line graph, which shows the closing price of Whole Foods Market stock each day from August \(16,2010\), to August \(20,2010\). (TABLE CAN NOT COPY) a. Complete the following table. In the row labeled "Equation," for each day, write an equation that can be used to find the amount Whole Food Market's stock closing price changed from the previous day's closing price. Use the variable \(n\) to represent the change in closing price. Solve the equation and write the amount the price changed in the row labeled "Price change." (TABLE CAN NOT COPY) b. Which day had the greatest change in closing price?
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