reinout-g

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Usernamereinout-g
Score2353
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Questions 70
Answers 574

 #1
avatar+2353 
+5

Hi there,

 

there's lot's of answers to your question, but I think you're just looking for an example of a fraction that is in between these two.

 

Let me give you some help here. Suppose I have 1 pizza and I cut it in 4 equal slices then if I take 1 slice I'll have 1/4 pizza. However if I cut the pizza in 8 slices and take 2 slices I'll still have 1/4 pizza.

 

Now suppose I have 1 pizza and I cut it in 11 equal slices then if I take 1 slice I'll have 1/11 pizza. However if I cut the pizza in 22 slices and take 2 I'll still have 1/11 pizza. Therefore we can say 1/11 = 2/22.

 

With similar reasoning we can also say that 1/12 = 2/24 (try to think of this pizza experiment yourself.)

 

So the question 1/11<...<1/12 is actually the same question as  2/22<...<2/24.

 

Now, I can divide a pizza in 22 slices, I can divide a pizza in 24 slices, but I could also divide it in 23 slices (Don't ask me to actually do it though, it would be very cumbersome).

 

Now if I have 2 slices of the pizza that is cut in 22 pieces those pieces will be bigger than 2 slices of the pizza that is cut in 23 pieces and those will again be bigger than the 2 slices of the pizza that is cut in 24 pieces.

 

Therefore we can state 2/22<2/23<2/24.

 

Now remember that we found that 1/11=2/22 and 1/12=2/24.

 

Hence we can also write 1/11<2/23<1/12.

 

And there you have an answer to your question, note however that it is not the only answer.

 

Suppose I cut the pizza in 44 slices and take 4 for myself I'd also have 1/11th pizza. 

 

Similarly as above we can therefore also reason that 4/44<4/45<4/46<4/47<4/48 and therefore that

 

1/11<4/45<4/46<4/47<1/12 (try to follow the reasoning above and do this for yourself.)

 

 

I hope this helped cool

Reinout

reinout-g Sep 7, 2015
 #1
avatar+2353 
+10

Let's first take a look at what happens in the first years.

Emily starts out with a salary of $25,000

After one year her salary has increased with 6%.

This means her salary is 106% of what it was last year.

Therefore her salary is now

$$(\frac{106}{100})\times\$25,000 = 1.06 \times\$25,000 = \$26,500$$

One year later her salary again increases with 6%

The means her salary is again 106% of what it was last year

Therefore her salary is now 

$$(\frac{106}{100})\times\$26,500 = 1.06 \times \$26,500 = \$28,090$$

 

Now suppose we had the same situation and we wanted to know what her salary was after 2 years.

Instead of first calculating her salary after one year we could also have immediately calculated

$$(\text{salary after two years}) = 1.06 \times (\text{salary after one year}) = 1.06 \times 1.06 \times (\text{starting salary}) = (1.06)^2 \times(\text{starting salary}) = (1.06)^2 \times \$25,000 = \$28,090$$

 

Similarly, we could show that for 3 years we have that 

$$(\text{salary after three years}) = 1.06^3 (\text{starting salary}) = 1.06^3 \times \$25,000$$

and that for x years we have that

$$(\text{salary after x years}) = 1.06^x = (\text{starting salary})$$

 

Now, if we rewrite the exercise as an equation we need to solve

$$1.06^x \times \$25,000 = 2 \times \$25,000$$

dividing both sides by 25000 gives us

$$1.06^x = 2$$

 

Now, I'm not sure what your level is so I'll give you two options

option 1 (the easy option (for whole years)):

Start with 1 and keep multiplying by 1.06 until the value is bigger or equal to 2.

The number of times you multiplied is the value of x.

To illustrate, this gives

$$\begin{array}{lcl}
1*1.06 = 1.06\\
1*1.06*1.06 = 1.06^2 = 1.1236\\
1*1.06*1.06*1.06 = 1.06^3 = 1.191016\\
(...)\\
1.06^{11} = 1.898298558\\
1.06^{12} = 2.012196472\\
\end{array}$$

Hence after 12 years, her income has doubled.

 

Option 2 (the advanced option (for partial years)):

This option makes use of logarithms. If you've never heard of logarithms, this method is not for you.

 

We indicate the natural logarithm as $$ln() = {}^elog()$$

We have that

$$\begin{array}{lcl}
1.06^x = 2\\
e^{ln(1.06^x)} = 2\\
e^{x \times ln(1.06)} = 2\\
x \times ln(1.06) = ln(2)\\
x = \frac{ln(2)}{ln(1.06)} \approx 11.8957
\end{array}$$

 Hence the answer is approximately 11.8957 years

Reinout 

reinout-g Apr 8, 2015
 
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