It seems like the main point being made here is that the expressions "a + 0.07a" and "1.07a" are equivalent because they represent the same calculation. The first expression breaks down the calculation into two parts - the initial cost of the book (a) and the sales tax (0.07a). The second expression combines these two parts into one by multiplying the initial cost by 1.07, which represents a 7% increase.
It is important to understand this equivalence because it allows us to simplify calculations and make connections between decimals and percentages. It shows that increasing a value by a certain percentage can be represented by multiplying it by a decimal equivalent.
Consider the following linear expressions:
a + 0.07a1.07a
What do these two linear expressions have in common? They each include the variable a , and they each have a number that has a decimal component of seven hundredths. The expressions are also equivalent. Remember that the variable a is really short for the expression 1a . Adding 1 and 0.07 gives 1.07, so the two expressions are indeed equivalent. The first expression can be thought of as “a certain amount a added to 7% of a .” The second expression can be thought of as “1.07 multiplied by a .”
Use the 21st century skill of managing goals and time to check over what you’ve learned and to be sure it makes sense.
This equivalence can be used to demonstrate something important about decimals and their relationship to percentages. A real-world context for these problems might be: A book costs a dollars. Sales tax is 7%. What expression gives the total cost of the book? The first expression steps this problem out into parts:
The first part, a , tells what you will pay in dollars for the book exclusive of the tax.
The second part, 0.07a , gives the amount in dollars of the sales tax.
The second expression folds these two amounts together into one. The fact that a+0.07a is equal to 1.07a shows that increasing a number by 7% is the same as multiplying it by 1.07 .
Again, before going on, use the skill of managing time and goals to make sure you are understanding the material.
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