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Chapter 1: Problem 35

Divide and then check by multiplying. $$6 \longdiv { 1 4 2 1 }$$

Short Answer

Expert verified
Quotient: 236, Remainder: 5. Verified by multiplication.

Step by step solution

01

Understand the Problem

The division problem \(6 \longdiv {1421}\) means we want to divide 1421 by 6 and find how many times 6 can fit into 1421.
02

Perform the Division

Start dividing from the leftmost digit:- **First Digit:** 6 fits into 14 two times (\(2 \times 6 = 12\)), subtract to get 2.- **Bring Down the Next Digit:** Bring down the 2 to make it 22.- **Second Digit:** 6 fits into 22 three times (\(3 \times 6 = 18\)), subtract to get 4.- **Bring Down the Next Digit:** Bring down the 1 to make it 41.- **Third Digit:** 6 fits into 41 six times (\(6 \times 6 = 36\)), subtract to get 5.So, the quotient is 236 and there is a remainder of 5.
03

Check by Multiplying

Multiply the quotient 236 by the divisor 6 and add the remainder 5:- \(236 \times 6 = 1416\)- Add the remainder: \(1416 + 5 = 1421\)Since the product plus the remainder equals the original dividend, the division is verified as correct.

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Key Concepts

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

Division with Remainder
When you divide numbers and don't end up with a perfect fit, you'll have something left over. That's what we call a remainder. Think of it like sharing cookies! If you have 22 cookies and share them with 4 friends, each gets 5 cookies and you're left with 2 cookies. Those 2 are your remainder.
In our example, we divided 1421 by 6. After doing the math, 6 fits into 1421 as much as possible, but there's still a little left, specifically 5. We write this as:
  • Quotient = how many times the number fits completely
  • Remainder = what's left over after that
So here, our quotient is 236, and the remainder is 5. Simple, right?
Quotient and Remainder
The quotient and remainder are essential parts of any division. Think of them as two outcomes from a single action. The quotient tells us the number of whole groups we can make, while the remainder tells us what's left at the end.
Using 1421 divided by 6, we found:
  • Quotient = 236
  • Remainder = 5
In the process of long division, each step calculates your partial results:
  • First, find how many times 6 fits into the first part of the number, 14, giving you 2.
  • This process continues with the next part, 22, resulting in 3.
  • And finally, for 41, giving you 6.
Putting those pieces together gives you 236 with a remainder of 5, and verifies our initial division.
Checking Division with Multiplication
Once you've completed your division, it's important to verify your answer. This ensures that no mistakes were made. You can double-check any division using multiplication and addition.
Here's how you can check:
  • Multiply the quotient by the divisor. This gives you back most of your original number.
  • Add the remainder to that product. If done correctly, this sum will match your original dividend.
Let's apply this to our example:
  • Multiply the quotient (236) by the divisor (6): \(236 \times 6 = 1416\).
  • Add the remainder (5): \(1416 + 5 = 1421\).
The final result, 1421, matches our original number, so we know everything checks out!

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