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brain puzzler

March 2018 Brain Puzzler Solution

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Q: Leaving at 8:00 AM, if Ms. Brown drives at an average speed of 40 miles per hour, she will be late by 3 minutes. If she drives at an average speed of 60 miles per hour, she will be early by 3 minutes.  How many miles per hour does Ms. Brown need to drive to get to work exactly on time?

A: 48 miles per hour

There is a 6 minute difference between the fast (60 mph) speed and the slow (40 mph) speed.  This 6 minutes is 1/10 of an hour.
Distance = rate x time.  The two distances are equal so:
We can figure out how long it would take at 60 mph and go from there.
40 (t + .1) = 60 t
40t  + 4 = 60 t
4 = 20 t
4/20 = t = 1/5 hour
So it will take 1/5 hour at 60 mph which is 12 miles. (1/5 hour = 12 minutes)
It will take abit longer at 40 mph:
1/5 hour + 1/10 hour =
.2 hour + .1 hour = .3 hour at 40 mph  which is also 12 miles. (.3 hour is 18 minutes).
If Ms. Brown was exactly on time it would take her 15 minutes to go the 12 miles.
12 miles in 15 min = 48 mph.

brain puzzler

February 2018 Brain Puzzler Solution

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Q: For a calendar, with two six sided dice make all possible monthly dates using two digits  (from 01 to 31) by writing one digit (from 0, 1, 2, …, 8, 9) on each face of the two dice.
A: It does not seem like it can be done — but…the Eureka part of the answer is:
the 9 is the 6 upside and vice versa!
Dice 1: 0 1 2 4 5 7 and  Dice 2: 0 1 2 3 8 9 (9 is also 6)
or
Dice 1 : 0 1 2 3 4 5 and Dice 2 : 0 1 2 6 7 8 (6 is also 9)

Common Core Algebra I, Regents, test prep

Get the Math and the Points Jan 2018 CC Alg I #2

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Jan 2018 CC Alg I 2
Two questions to ask:
1. Does V(x) grow or decay?
2. When putting 4(0.65)^x vs 4(1.35)^x into the Y= of the TI-83/4, does the table look like the one above?
Hopefully the student will know that the number in parentheses (the growth/decay factor) determines if it is growth or decay but through the graphing calculator, they can figure it out, get the right answer and earn 2 points!

brain puzzler

January 2018 Brain Puzzler Solution

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Q: What is the sum of the digits of the square of $111,111,111$?

A: 81
Using the standard multiplication algorithm, $111,111,111^2=12,345,678,987,654,321,$whose digit sum is 81.

Or by looking at the pattern:

$11^2 = 121 \\ 111^2 = 12321 \\ 1111^2 = 1234321$

$111,111,111^2=12,345,678,987,654,321$ whose digit sum is 81
There is a shortcut to adding these digits
Reading from left to right, we can add the first 8 digits, 1 through 8 by making 4 pairs of 9 (1 + 8, 2 + 7, 3 + 6 and 4 + 5) making 36
Reading from right to left, we can add the last 8 digits (1- 8) the same way making 36.
Leaving the middle digit of 9.

36 + 36 + 9 = 81

Common Core Algebra I, Regents, test prep

Get the Math and the Points! CC Alg I June 2017 Regents #10

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June 2017 CC Alg I 10Can use the TI-83/4 to figure this one out!
The zeros are the values of x that make f(x) or y equal to zero.
Answer (1) looks like this:  only 1 zero at x = -3 which can be seen on both the graph and the table
Aug 2017 CC Alg I 10a EqAug 2017 CC Alg I 10aAug 2017 CC Alg I 10a Graph

Answer (2) looks like this with 3 zeros but only one of them is an integer value (both the table and the graph show that (-4,0) is a point
Aug 2017 CC Alg I 10b yAug 2017 CC Alg I 10b tableAug 2017 CC Alg I 10b graph

Let’s try answer (3):

Aug 2017 CC Alg I 10c EqAug 2017 CC Alg I 10c tableAug 2017 CC Alg I 10c graph
Here we can see that on the table when x = -3 y = 0 and also when x = 0 and x = 4, y is also equal to zero.  Looking at the graph we can see that the 3 x-intercepts are integer values of -3, 0 and 4.  That’s it!

Pay close attention to the signs!! We need to look for the factors with the opposite sign to ‘zero it out’!  This can be a bit counterintuitive as when -3 is a solution then the factor is (x + 3), as -3 makes x + 3 = 0.  When 4 is a solution, the factor must be (x – 4)  rather than  (x+4) as plugging in 4 would not result in zero.