The colour of some clothing fades overtime when washed. Suppose a pair of jeansfades by 5% with each washing. To the nearest wash, how many washes would it take so that only 25% of the original colour remains in the jeans?

Question

The colour of some clothing fades overtime when washed. Suppose a pair of jeansfades by 5% with each washing. To the nearest wash, how many washes would it take so that only 25% of the original colour remains in the jeans?

1. first term =
2. r =
3. tn =
4. n =

I know that tn is 25 because that's the term that they're asking for but I cna't figure out the "r"... I thought it was 19/20 because when 5% fades, 95% of the color is left but I got it wrong.

Answer 1

you want:

.95^n = .25
take log of both sides

nlog.95 = log.25
n = log.25/log.95 = 27.0268..

= appr 27 washings

or

start with 100% colour
after 1 washing ---- 95% left
after 2 washings --- .95(.95) left
after 3 washings --- .95(.95)^2 left
....
after n washings ------ .95(.95)(n-1) left , should be .25

so .95(.95)^(n-1) = .25
.95^n = .25 , which is the same equation I started with

checking my answer of 27 washings
amount left over = .95(.95)^26 = .25034
or appr 25%

Answer 2

don't make confusing, because if u look at the sequence its 100, 95, 90.25, 85.7375

first term is 100 this is the original color and second term is 95 and this is second term but first hand wash if you follow the pattern then for term three number of hand wash is two and for term four number number of hand wash three and so on . hence for this situation 10 washings for 11th term : t11=100(.95)^10=59.87
and we have happy ending

Answer 3

To find the value of "r," we need to determine the ratio of the faded color to the original color after each wash. Since the jeans fade by 5% with each washing, the ratio of faded color to original color after each wash is 100% - 5% = 95%.

Now, let's represent the ratio of faded color to original color after each wash as a decimal fraction:

r = 95% = 0.95

The first term (original color) is 100%, which is equivalent to 1.

Using the formula tn = a * r^(n-1), where tn represents the nth term, "a" is the first term, "r" is the common ratio, and "n" is the number of washes, we can calculate the number of washes required for only 25% of the original color to remain in the jeans.

Given that tn = 25%, we substitute the values into the formula:

0.25 = 1 * 0.95^(n-1)

Simplifying the equation:

0.25 = 0.95^(n-1)

Now, let's solve for "n" by taking the logarithm base 0.95 on both sides of the equation.

log(0.25) = log(0.95^(n-1))

Using logarithmic properties, we can rewrite the equation as:

(n-1) * log(0.95) = log(0.25)

Now, divide both sides of the equation by log(0.95) to get the value of (n-1):

(n-1) = log(0.25) / log(0.95)

Finally, adding 1 to both sides of the equation, we can find the value of "n":

n = log(0.25) / log(0.95) + 1

Evaluating this expression will give us the nearest whole number for the number of washes required for only 25% of the original color to remain in the jeans.

Answer 4

To solve this problem using a geometric sequence, we need to determine the common ratio (r) first.

The fade percentage for each wash is given as 5%, which means that 95% of the color remains after each wash. So, the ratio of color remaining after each wash is 0.95 (since 95% can also be written as 0.95).

Now let's solve the questions:

1. first term (a1): We are not given the initial color percentage, so we cannot determine the first term from the information provided. We will assume it is 100%, so a1 = 1 (since 100% can also be written as 1).

2. common ratio (r): As mentioned earlier, the ratio of color remaining after each wash is 0.95, so r = 0.95.

3. tn: The term we are interested in is where only 25% of the original color remains. We'll call this term tn. Therefore, tn = 0.25 (since 25% can also be written as 0.25).

4. n: We need to find the number of washes it would take to reach tn. Since we start at a1 and progress to tn, we can use the formula tn = a1 * (r^(n-1)).

Substituting the values we have:
0.25 = 1 * (0.95^(n-1))

Simplifying the equation:
0.95^(n-1) = 0.25

To solve for n, take the logarithm (base 0.95) of both sides:
log base 0.95 (0.95^(n-1)) = log base 0.95 (0.25)

(n-1) = log base 0.95 (0.25)
n = log base 0.95 (0.25) + 1

Using a calculator or software to evaluate, we find that n is approximately equal to 20.9 washes. Since we cannot have a fraction of a wash, we round up to the nearest whole number.

Therefore, it would take approximately 21 washes for only 25% of the original color to remain in the jeans.