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Chapter 2: Problem 33

Find each product or quotient. $$ 84 \div 15 $$

Short Answer

Expert verified
The quotient of 84 divided by 15 is 5 with a remainder of 9.

Step by step solution

01

Set Up the Division

We have the division problem \( 84 \div 15 \). This means we need to find out how many times 15 fits into 84 completely (without exceeding it) and determine the remainder, if any.
02

Division Estimate

Estimate how many times 15 goes into the first number in 84 (which is 8) and then consider 84 as a whole. Since 15 is greater than 8, we consider 84. Use integer division to estimate the quotient by dividing 84 by 15 to get approximately 5. This means 15 fits into 84 five times without going over.
03

Calculate the Quotient

Multiply 5 (our estimated quotient) by 15 to see how much of 84 we can cover: \( 5 \times 15 = 75 \).

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Key Concepts

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

integer division
Integer division involves dividing two whole numbers to obtain another whole number as a result. This type of division does not deal with fractions or decimal points. Instead, it only considers how many times the divisor can fit into the dividend fully and discards any fractional part of the quotient.
When performing integer division, we often use the syntax \( a \div b \), where \( a \) is the dividend and \( b \) is the divisor. The result will always be an integer.
The key steps in integer division include:
  • Determine how many times the divisor can fit into the dividend fully.
  • Ignore any remainder that does not allow a full fit of the divisor.
These principles ensure that integer division remains simple and helps in understanding more about divisibility between numbers.
quotient
The quotient is the result of a division operation, showing how many times a divisor can fit into the dividend completely. In the example \( 84 \div 15 \), the quotient is 5.
To find the quotient, follow these steps:
  • Divide the total or dividend by the divisor. Consider how many whole times it goes into the dividend without exceeding.
  • In this case, 15 goes into 84 five times, which makes the quotient 5.
The quotient is an essential part of division as it provides a whole number answer to the division problem, telling us how many full groups we can obtain.
remainder
After calculating the quotient, there might be a leftover part of the dividend that cannot be fully divided by the divisor. This part is called the remainder.
The remainder shows what is left after dividing the dividend completely by the quotient multiplied by the divisor.
Returning to our example, once we calculate \( 84 \div 15 \) and find the quotient to be 5, we multiply:
\( 5 \times 15 = 75 \).
Subtract 75 from the dividend 84:
\( 84 - 75 = 9 \).
So, the remainder is 9, because there isn鈥檛 enough left to form another complete group of 15.
The remainder is always smaller than the divisor and acts as the leftover piece of the dividend in the division operation.

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