Solution
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To calculate the product of 4 times 9, you will need to perform a multiplication operation. Multiplication is one of the four elementary mathematical operations of arithmetic; with the others being addition, subtraction, and division. The multiplication of two whole numbers is equivalent to adding as many copies of one of them, the multiplicand, as the quantity of the other one, the multiplier. Both numbers can be referred to as factors.
To calculate the product of 4 times 9, you will need to perform a multiplication operation. Multiplication is one of the four elementary mathematical operations of arithmetic; with the others being addition, subtraction, and division. The multiplication of two whole numbers is equivalent to adding as many copies of one of them, the multiplicand, as the quantity of the other one, the multiplier. Both numbers can be referred to as factors.
Here is how you would calculate the product step by step:
Step 1: Identify the multiplicand and the multiplier. In this case, 4 is the multiplicand, and 9 is the multiplier.
Step 2: Set up the multiplication expression. This is done by writing the two numbers with the multiplication sign between them. The multiplication sign can be represented by an 'x', an asterisk '*', or a centered dot '·'. We will use the asterisk for this example.
Step 3: Calculate the product. To find the product of 4 and 9, you can add the number 4 to itself 9 times, or you can recall the multiplication fact from memory if you know your times tables.
Step 4: If you are adding 4 to itself 9 times, it would look like this:
When you add up all the fours, you get:
Alternatively, if you recall the multiplication fact from memory, you would know that:
Therefore, the product of 4 times 9 is 36.
Here is how you would calculate the product step by step:
Step 1: Identify the multiplicand and the multiplier. In this case, 4 is the multiplicand, and 9 is the multiplier.
Step 2: Set up the multiplication expression. This is done by writing the two numbers with the multiplication sign between them. The multiplication sign can be represented by an 'x', an asterisk '*', or a centered dot '·'. We will use the asterisk for this example.
Step 3: Calculate the product. To find the product of 4 and 9, you can add the number 4 to itself 9 times, or you can recall the multiplication fact from memory if you know your times tables.
Step 4: If you are adding 4 to itself 9 times, it would look like this:
When you add up all the fours, you get:
Alternatively, if you recall the multiplication fact from memory, you would know that:
Therefore, the product of 4 times 9 is 36.
Exercise 10 - Time for a car to lose half of its value
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Exercise 10 - Time for a car to lose half of its value
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This problem talks about a car depreciating in value exponentially,
and this makes it an exponential decay problem.
When something loses value,
there's more than 1 formula we could use.
If it was a problem where we were told how many percent it loses each year,
then we would use this formula which is like the formula for interest,
except for the minus instead of a plus.
But we're not going to use this one because we
don't talk about the percentage loss each year,
and in fact, we don't even need to figure out what percent it is.
What we do, is instead of this 1 minus r/100,
we use a different letter,
say R to represent all this.
This is the depreciation factor,
this is what you multiply the value by each year to see what the new value is.
If you do this replacement,
what you get is the formula that we are going to use,
and at this formula,
normally P is for principal and A is for amount because it's used in interest problems.
But in our case, P is the initial value and
A would be the final value after a period of t years.
Let's first of all look at the part where it loses 25 percent of its value.
As a side exercise,
I'd like to ask if something has value P and it
loses 25 percent of its value,
then really it's lost 25 percent of P. In other words,
it's P minus 0.25P,
and that comes out to 0.75P.
That would be the value that was originally value P,
and if it loses 25 percent,
it's now only 0.75P,
which you can see it's 3/4 of its original value.
Now we can say that 0.75P,
that's the A, the value after,
will equal the original value P times R,
and we know that this is 4 years.
Now, the P cancels and that's good because now we just have 1 variable R,
we can say that if R^4 is 0.75,
then R is the 4th root of 0.75.
I won't calculate it just yet,
I'll just point out that it might be more useful to have it in the form 0.75^1/4,
4th root of a quarter, whatever it turns out to be more convenient for us.
That relates to the 25 percent loss of 4 years.
Now let's look at the 50 percent loss.
We don't know after how many years,
the how many years,
I'll call that t, that's the unknown.
We did the same thing,
but with 50% instead of 25% we'd
get P minus 50%(P),
and it would come out to be 0.5P.
We get a different formula for this,
is that 0.5P,
which is the amount after losing 50%,
is the original amount times R to the power of,
well, that's what we're looking for,
t is our unknown,
but we do know that it's the same R from here.
We can write this as P times 0.75,
I'll use this from the power of a quarter,
and all this to the power of t, and that's 0.5P.
Continuing, before the P cancels,
and what we're left with is that,
I'll reverse the sides.
I've got 0.75^1/4 times t,
using the rules of exponents,
I can write this in fact as t/4=0.5.
Now I will take the logarithm of both sides,
I'll take base 10,
any base will do,
like base 10 or just log,
so I get t/4, log0.75=log0.5.
Now we can get what t equals,
t equals just multiplied by 4,
so we've got 4log0.5 and then divide by this,
so it's over log0.75.
Now we have a numerical expression for t,
we just have to calculate it using the calculator.
Let's see, this is what I make it,
and there's too many decimal places here.
Let's round it,
and we'll say it's approximately,
I'll round it to 2 decimal places,
I just have to write the word years,
and that is the answer.
If it loses a quarter of its value in 4 years,
to lose 1/2 of its value takes 9.64 years, and we're done.
This video discusses the exponential decay of a car's value. It explains how to use the formula P(1-R)^t to calculate the value of a car after a certain period of time, given the initial value and the depreciation factor R. It also explains how to calculate the depreciation factor R when given the initial value and the final value after a certain period of time. As an example, the video explains how to calculate the time it takes for a car to lose 50% of its value, given that it loses 25% of its value in 4 years. The answer is 9.64 years.
• Explains exponential decay of car's value
• Formula P(1-R)^t to calculate value of car after certain period of time
• Calculate depreciation factor R when given initial value and final value
• Example: Calculate time it takes for car to lose 50% of its value, given it loses 25% of its value in 4 years
• Answer: 9.64 years
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