Two weeks before an examination day, a librarian monitored the number of students who visited the library. Then they compared it with the number of visits made in an average week. Approximately 2×10^3
visits were reflected in their logbook during this two-week period, while only about 5×10^2
were reflected on an average week. How many times higher is the number of library visits before examination day compared to the library visits in an average week? Your answer should be a whole number.
To find the number of times higher the number of library visits before examination day is compared to the number of visits in an average week, we need to divide the number of visits before examination day by the number of visits in an average week.
The number of visits before examination day is approximately 2×10^3.
The number of visits in an average week is about 5×10^2.
Dividing the number of visits before examination day by the number of visits in an average week:
(2×10^3) / (5×10^2) = (2/5) * (10^3 / 10^2) = 0.4 * (10^(3-2)) = 0.4 * 10^1 = 0.4 * 10 = 4
Therefore, the number of library visits before examination day is 4 times higher than the number of visits in an average week.
To solve this problem, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment
P = the principal amount (the monthly deposit)
r = the annual interest rate (in decimal form)
n = the number of times that interest is compounded per year
t = the number of years
In this case, the principal amount P is $100 per month, the annual interest rate r is 5% or 0.05, the number of times interest is compounded per year n is 12, and the number of years t is 10.
Plugging in these values, we get:
15,528.23 = 100(1 + 0.05/12)^(12*10)
Dividing both sides by 100, we get:
155.2823 = (1 + 0.05/12)^(120)
Taking the natural logarithm of both sides, we get:
ln(155.2823) = ln((1 + 0.05/12)^(120))
Using the properties of logarithms, we can simplify the right side of the equation:
ln(155.2823) = 120 * ln(1 + 0.05/12)
Now, we can solve for ln(1 + 0.05/12) by using the Taylor series approximation:
ln(1 + x) ≈ x for small values of x
So, ln(1 + 0.05/12) ≈ 0.05/12
Plugging this approximation into our equation, we get:
ln(155.2823) ≈ 120 * (0.05/12)
ln(155.2823) ≈ 0.05
Finally, taking the exponential of both sides, we get:
e^(ln(155.2823)) ≈ e^(0.05)
155.2823 ≈ 1.05127
Rounded to the nearest place value, the future value of the investment is $155.28.
Expressing $155.28 in the form of a×10^n, where a is a single digit:
$155.28 = 1.5528 × 100
Therefore, the amount $15,528.23 can be expressed approximately as 1.5528 × 10^4.
Round to the nearest place value, then express the amount $15,528.23 in the form of a×10^n where a is single digit
Rounded to the nearest place value, the amount $15,528.23 becomes $15,528.
To express $15,528 in the form of a×10^n where a is a single digit, we need to move the decimal point one place to the left to make it a value between 1 and 10. We also need to keep track of the decimal place shift and adjust the exponent of 10 accordingly.
$15,528 = $1.5528 × 10^4
Therefore, the amount $15,528.23 can be expressed as approximately 1.5528 × 10^4.
To find out how many times higher the number of library visits before the examination day is compared to the library visits in an average week, we need to calculate the ratio of the number of visits before the examination day to the number of visits in an average week.
Number of visits before examination day = 2×10^3
Number of visits in an average week = 5×10^2
Ratio = (Number of visits before examination day) / (Number of visits in an average week)
Ratio = (2×10^3) / (5×10^2)
To simplify, we can divide both the numerator and denominator by 10^2:
Ratio = (2×10^3) / (5×10^2)
Ratio = 2/5 × 10^3 / 10^2
Ratio = (2/5) × 10^(3-2)
Ratio = (2/5) × 10^1
Ratio = 2/5 × 10
Multiplying 2/5 with 10 gives us:
Ratio = (2/5) × 10
Ratio = 2 × (10/5)
Ratio = 2 × 2
Ratio = 4
Hence, the number of library visits before the examination day is 4 times higher than the number of visits in an average week.
To find out how many times higher the number of library visits before the examination day is compared to the number of visits on an average week, we need to divide the number of visits before the examination by the number of visits on an average week.
Let's calculate the number of visits before the examination day:
2 × 10^3 visits
Now, let's calculate the number of visits on an average week:
5 × 10^2 visits
To find out how many times higher the number of visits before the examination day is, we divide the number of visits before the examination by the number of visits on an average week:
(2 × 10^3) / (5 × 10^2) = (2/5) × (10^3/10^2) = (2/5) × 10^(3-2) = (2/5) × 10^1 = 2/5 × 10 = 0.4 × 10 = 4
Therefore, the number of library visits before the examination day is 4 times higher compared to the number of visits on an average week.
It takes 10 years for a $100 monthly bank deposit to amount to $15,528.23 when invested at 5% interest compounded monthly. Round to the nearest place value, then express the amount $15,528.23 in the form of a×10^n
where a
is single digit.