If multiplying by 1/4 makes a positive number smaller, then what does dividing by 1/4 do to the value of the number? Explain your reasoning.
Dividing by 1/4 is the same as multiplying by its reciprocal, which is 4.
When multiplying a positive number by 1/4, the result becomes smaller because we are taking only a fraction of the original value.
Therefore, if we multiply the smaller result by 4, we are essentially canceling out the effect of multiplying by 1/4. Since we are left with the original value, dividing by 1/4 restores the number back to its original size.
If multiplying by 1 does not change the value of a number, then what effect does multiplying by 2/2 have? Explain your reasoning.
Multiplying by 1 does not change the value of a number because any number multiplied by 1 remains the same.
Multiplying by 2/2 is the same as multiplying by 1, since 2/2 simplifies to 1.
Therefore, multiplying by 2/2 also does not change the value of the number. It has no effect because it is equivalent to multiplying by 1, which leaves the number unchanged.
Dividing by 1/4 is the same as multiplying by its reciprocal, which is 4/1 or 4. So, if multiplying by 1/4 makes a positive number smaller, then dividing by 1/4 would make it larger.
To understand this reasoning, let's consider an example. Suppose we have the number 8. When we multiply it by 1/4, we get 8 * 1/4 = 2. So, multiplying by 1/4 makes 8 smaller and reduces its value.
Now, let's divide 8 by 1/4, which is the same as multiplying it by 4. 8 * 4 = 32. We can see that dividing by 1/4 increases the value of 8 and makes it larger.
In general, when you multiply by a fraction that is less than 1, such as 1/4, it reduces the value of the number. Conversely, when you divide by a fraction that is less than 1, it increases the value of the number. This is because dividing by a fraction less than 1 is equivalent to multiplying by its reciprocal, which is greater than 1.
So, if multiplying by 1/4 makes a positive number smaller, then dividing by 1/4 makes it larger.