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Chapter 1: Problem 31

Show that \(6 \div 3=2\) but \(3 \div 6 \neq 2\) by using multiplication to check.

Short Answer

Expert verified
6 梅 3 = 2 but 3 梅 6 鈮� 2 by checking via multiplication.

Step by step solution

01

Understand Division as Multiplication

To check a division problem using multiplication, multiply the quotient (result of division) by the divisor. The product should be equal to the dividend (what you are dividing).
02

Verify that 6 梅 3 = 2

Perform the division to get the quotient: 6 梅 3 = 2. Now, multiply the quotient by the divisor: 2 脳 3 = 6. Since the product (6) is equal to the dividend (6), the division is correct.
03

Verify that 3 梅 6 鈮� 2

Perform the division to see what the quotient is: 3 梅 6 = 0.5. Now, multiply the quotient by the divisor: 0.5 脳 6 = 3. However, 2 脳 6 = 12, which is not equal to the dividend (3). Thus, 3 梅 6 is not equal to 2.

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Key Concepts

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

Division as Multiplication
Division can always be checked using multiplication. Here鈥檚 how it works: you take the quotient (the answer you get from dividing) and multiply it by the divisor (the number you are dividing by). If you get back to the dividend (the number you started with), your division is correct.

For example, if you have a problem like 6 梅 3 = 2, you can check it by reversing the operation: multiply the quotient, 2, by the divisor, 3. If the result is 6, then the division was done correctly.

Always remember:
  • The divisor is the number you divide by.
  • The dividend is the number being divided.
  • The quotient is the answer of the division.
Quotient
The quotient is simply the result you get after performing a division. In the problem 6 梅 3 = 2, the quotient is 2.

You should always remember that the quotient times the divisor should give back the dividend if the division was done correctly. For example:
  • In 6 梅 3 = 2, 2 is the quotient.
  • Checking: 2 脳 3 should equal 6.
In cases where the quotient is not as obvious, like 3 梅 6, the quotient becomes a fraction or decimal. Here, 3 梅 6 = 0.5.

Checking again: 0.5 脳 6 = 3. This confirms that the quotient of 3 梅 6 is correct.
Dividend
The dividend is the number you want to divide. In the given exercises, the dividends are 6 and 3.

The dividend is the starting point for your division, and using multiplication helps see if you end up at the same number, confirming the operation. Consider these steps:
  • 6 梅 3 gives a quotient of 2.
  • Checking: 2 脳 3 returns you to the dividend, 6.
For 3 梅 6:
  • The quotient is 0.5.
  • Checking: 0.5 脳 6 returns you to the dividend, 3.
In both cases, multiplication confirms the dividend when done correctly.
Divisor
The divisor is the number by which you divide the dividend. Divisors in this case are 3 and 6.

To verify the accuracy of any division, multiply the quotient by the divisor and see if it equals the dividend.

Examples:
  • In 6 梅 3 = 2, the divisor is 3.
  • Checking: 2 脳 3 = 6, confirming the divisor.
Similarly,
  • For 3 梅 6, the divisor is 6.
  • Checking: 0.5 脳 6 = 3.
In each situation, the divisor plays a crucial role in verifying the division through multiplication.

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