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Chapter 1: Problem 68

Which of the following four expressions differs from the remaining three? $$ 3 \backslash \text { longdiv } 15,3 \div 15, \frac{15}{3}, 15 \div 3 $$

Short Answer

Expert verified
The different expression is 3 梅 15.

Step by step solution

01

Calculate 3 \ longdiv 15

This expression represents how many times 3 goes into 15. Mathematically, it is equivalent to evaluating the division of 15 by 3. Hence, \(3 \ \text{longdiv} \ 15\) equals to 0 with a remainder of 3.
02

Calculate 3 梅 15

This is a straightforward division of the number 3 by 15. Using division, we find that \(3 \div 15 = 0.2\).
03

Calculate \(\frac{15}{3}\)

This is a straight division where 15 is divided by 3. Thus, \(\frac{15}{3}\) results in 5.
04

Calculate 15 梅 3

Again, this division directly implies dividing 15 by 3. As a result, \(15 \div 3 = 5\).
05

Identify the differing expression

Comparing the results, we have 0 from Step 1, 0.2 from Step 2, and 5 from Steps 3 and 4. The expression \(3 \div 15\) results in 0.2, which is different from the others, which provide whole numbers either as results or results suggest otherwise.

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Key Concepts

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

Long Division
Long division is a method of dividing large numbers. It breaks down the complex division process into a series of simpler steps. By using long division, you tackle the problem step by step, performing a series of subtractions until the division is complete to find the quotient and remainder.
  • First, write the dividend under the division symbol and the divisor outside of it.
  • Check how many times the divisor fits into the first digits of the dividend.
  • Write the result above the division symbol as the first digit of the quotient.
  • Multiply the divisor by this digit and subtract from the dividend portion.
  • Bring down the next digit and repeat the process until all digits are used.
Understanding this is crucial because in problems like 3 \(\backslash\) longdiv 15, you find how many times 3 fits into 15. Here, long division tells us it fits 0 times, leaving a remainder of 3, showing that the partial product of the division doesn't fully equal the dividend, indicating more steps or a remainder as needed.
Fraction Division
Fraction division involves dividing two fractions or dividing a fraction by a whole number. It directly relates to understanding ratios and proportions. Here's the key process:
  • Change the division sign to multiplication.
  • Take the reciprocal of the divisor. Essentially flip the numerator and denominator.
  • Multiply the fractions.
For example, let's look at dividing a simple number by a fraction. If we calculate the expression \( \frac{15}{3} \), we recognize it as 15 divided by 3 rather than 3 divided by 15. The result is a whole number, 5, because three whole parts fit into 15 exactly. Understanding these basic principles makes fraction division more intuitive and helps you solve similar math problems without confusion.
Decimal Division
Decimal division handles numbers with one or more decimal points, requiring care to ensure place value is maintained. Steps here help manage this complexity:
  • Align the decimal points of both numbers if both are decimals.
  • If only the dividend includes decimals, you can ignore placement and divide as whole numbers.
  • If the divisor is a decimal, adjust by multiplying both numbers by the power of ten needed to convert it to a whole number.
  • Perform the division as with integers, bringing the decimal point directly up into the result.
For instance, considering 3 梅 15 gives a result of 0.2 as you've divided using decimal points as a means to quantify how 15 parts fit into 3 fractions of a whole. The key idea is aligning the decimal positioning ensures precision in the result. Recognizing decimal division's nuances will enhance your skill in solving division problems that result in non-integer answers.

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Most popular questions from this chapter

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