Solutions to Some of the Impossible Problems

A3 Solve for x: 2^2x+7 = 8^x+1

Since 8=2³ let’s rewrite that as 2^2x+7 = 2^3(x+1)  

So 2x+7 = 3(x+1)

distribute the 3:  2x+7= 3x+3

take 2x from both sides 7=x+3

So x=4

Let’s check our answer. Replace x with 4: 2^2x+7 = 8^x+1   

2^2x4+7 = 8⁴⁺¹ 

So 2¹⁵ = 8⁵

Is that true? Without doing all the math let’s express 2¹⁵ as

(2x2x2)(2x2x2)(2x2x2)(2x2x2)(2x2x2)

Yep, that’s clearly the same as 8⁵

A5 How many integers have a square root between 8/3 and 15/4 ?

Squaring both sides of the sentence, that's really the same as asking 

"How many integers are between 64/9 and 225/16?"

or how many integers are between 7.1 and 14.1?

Well, 8, 9, 10, 11, 12, 13, 14. Seven of them.

B3. What value of m makes the equation true?

4 - m    =       m       

4+m         m+m+m+m

That's 4 - m    =   m  

           4+m         4m

That's 4 - m    =   1 

           4+m          4

Or 4(4 - m) = 4 + m

Thus 16 - 4m = 4 +m

Or 16 = 4 + 5m

Or  12 =5m

Or 12/5 = m

B4.  Mr. Olson decided to sell his old washer and dryer. He sold each appliance for $50, which was 25% more than what he expect to get for the washer, and 20% less than what he expected to get for the dryer.  In total, how much was Mr. Olson expecting to make for the sale of his appliances?

Let W= what he expected to get for the washer and D= what he expected to get for the dryer.

W+ .25W = 50    D-.20D = 50

1.25W = 50         .80D = 50

W = 40                  8D = 500

                              D=62.50

So what was the question? Oh, he expectd W+D = 40 + 62.50 = $102.50

T4. The price of a stock went up 20% then down 20% before it sold for $48. What was the original price of the stock?

Let P be the original price.  So P(1.20)(.80)= $48

Divide both sides by .8 to get P(1.20)= $60

Now divide by (1.20) to get P = $50

Let's check Price $50 up 20% - that's up $10 to $60. Now down 20%. Down 10% would be down $6 so 20% is down 12... it was at 60 so it's $48. Yep.

And finally my favorite impossible problem (drumroll please)

T9. A 5-digit number will be turned into a six digit number by adding a 1 to the beginning of end of the number. If 1 is added to the end the resulting 5-digit number is triple what it would be if the 1 had been added to the beginning. What's the original 5-digit number?

Let n be the 5-digit number. If I add 1 to the end of the number, that's like multiplying by 10 and adding 1. If I add 1 to the beginning that's like adding 100,000. So let's write the equation.

10n + 1 = 3 (100,000 + n)

10n + 1 = 300,000 + 3n

7n + 1 = 300,000

7n = 299,999

n = 42857

I'll let you multiply 3 x 142857 to verify the magical answer.