Rate, Time, and Distance Question

Question: Harder SAT question: Esther drove to work in the morning at an average speed of 45 miles per hour. She returned home in the evening along the same route and averaged 30 miles per hour. If Esther spent a total of one hour commuting to and from work, how many miles did Esther drive to work in the morning?

Solutions I:
Let t be the time it took Esther to drive to work and (1-t ) be the time it took him to return home.

Since rate times time = distance, we can set the equation as 45 t = 30 (1- t), 75t = 30 so
t = \(\frac{\Large{2}}{\Large{5}}\) Plug in t to the previous equation : 45 * \(\frac{\Large{2}}{\Large{5}}\) = 18 miles ,which is the answer.

Solution II: 
Let D be the distance from Esther's home to work. 

\(\frac{\Large{D}}{\Large{45}}\) +  \(\frac{\Large{D}}{\Large{30}}\) = 1 (hour)
Times 90 both sides to get rid of the denominator and you have 2D + 3D = 90 
D = 18 miles

Solution III:

The rate ratio between driving to work and returning home is 45 : 30 or 3 :  2.

Since rate and time are inversely related (rt = d), the time ratio between the two is 2 : 3. 
45*\(\frac{\Large{2}}{\Large{5}}\) * 1 (hour) = 18 miles

Solution IV: (from Varun in FL)
The average speed can be found by harmonic mean: \(\dfrac {2\times 45\times 30}{45+35}\) = 36 mph. She drives for a total of 1 hour, so she drives 36 miles. Since she drives the same distance both ways, the distance to work is \(\dfrac {36} {2}\)=18 miles.

Question 1991 Mathcounts National #28 : A man is running through a train tunnel. When he is \(\frac{\Large{2}}{\Large{5}}\) of the way through, he nears a train that is approaching the tunnel from behind him at a speed of 60 mph. Whether he runs ahead or runs back, he will reach an end of the tunnel at the same time the train reaches that end. At what rate, in miles per hour, is he running? (Assume he runs at a constant rate.)

Solution : When the man went back \(\frac{\Large{2}}{\Large{5}}\) of the length of the tunnel, the train is at the beginning of the tunnel. 

Same thing happened if the men went ahead \(\frac{\Large{2}}{\Large{5}}\) of the length of the tunnel. The time it took the train to run through the length of the tunnel, the man could only run \(\frac{\Large{1}}{\Large{5}}\) of the same distance. 

So the speed of the man is \(\frac{\Large{1}}{\Large{5}}\) of the train's speed, which is \(\frac{\Large{1}}{\Large{5}}\) * 60 or 12 mph.  










Question #3: Sally drives to her aunt's house, which is 84 miles away, at 40 miles per hour and comes back home at 60 miles per hour. What is her average speed? 
Solution I: 
To find average speed, you use total distance over total time it takes Sally to drive. 
It takes Sally \(\frac{\Large{84}}{\Large{40}}\) + \(\frac{\Large{84}}{\Large{60}}\), or total 3.5 hours 
to drive 84 * 2 = 168 miles. 
\(\frac{\Large{168}}{\Large{3.5}}\) = 48 mph

Solution II: 
Similar to dimensional change question, as long as the segments are constant, the distance Sally drives is extraneous. Thus, it's much easier to use \(\frac{\Large{2}}{\frac{\Large{1}}{\Large{40}} - \frac{\Large{1}}{\Large{60}}}\)or \(\frac{\Large{2*40*60}}{\Large{40+60}}\) = 48 mph


Question #4: If Sally drives to her work at 40 miles per hour, she will be 15 minutes late. If she drives to her work at 60 miles per hour, she will be 15 minutes early, what speed will she arrive at her work on time?  How far away is her office ? 
Solution I: 
Let t be the time (hpm) Sally drives to her work on time. According to the given, we can set up 
the following equation: 
40 (t + \(\frac{\Large{1}}{\Large{4}}\)) = 60 ( t- \(\frac{\Large{1}}{\Large{4}}\) )
40 t + 10 = 60 t - 15 ;      25 = 20t ;       t = 1.25
Plug in to get the distance as 40 (1.25 + 0.25) = 60 miles

To get the average, use total distance over total time she spent \(\rightarrow\) \(\frac{D}{T}\) = \(\frac{\Large{120}}{\Large{1.5 + 1}}\) = 48 mph

Solution II :
Again, have you noticed that if  both time (late or early) to arrive to the destination is the same, the easiest
method is to still use the harmonic mean. \(\rightarrow\) \(\frac{\Large{2*a * b}}{\Large{a + b}}\)
a and b are the rates. \(\frac{\Large{2*40*60}}{\Large{40+60}}\) = 48 mph
 

Unit digit, Tenth digit and Digit Sum

Word problems on unit digit, tenth digit or digit sum.

#1: How many digits are there in the positive integers 1 to 99 inclusive? 

Solution I:  From 1 to 9, there are 9 digits.
From 10 to 99, there are 99 - 10 + 1 or 99 - 9 = 90 two digit numbers. 90 x 2 = 180
Add them up and the answer is 189.

Solution II:  ___ There are 9  one digit numbers (from 1 to 9).
___ ___ There are 9 * 10 = 90 two digit numbers (You can't use "0" on the tenth digit but you
can use "0" on the unit digit.) 90 * 2 + 9 = 189

# 2: A book has 145 pages. How many digits are there if you start counting from page 1?

There are 189 digits from page 1 to 99. (See #1, solution I)
From 100 to 145, there are 145 - 100 + 1 or 145 - 99 = 46 three digit numbers.
189 + 46*3 = 327 digits.

#3: "A book has N pages, number the usual way, from 1 to N. The total number of digits in the page number is 930. How many pages does the book have"?  Similar to one Google interview question.

Read the questions and others here from the Wall Street Journal.

Solution I: 

930 - 189 (digits of the first 99 pages) =741
741 divided by 3 = 247. Careful since you are counting the three digit numbers from 100 if the book has N
pages N - 100 + 1 or N - 99 = 247. N = 346 pages.

Solution II: 

930 - 189 (total digits needed for the first 99 pages) = 741
741/3 = 247 (how far the three digit page numbers go).
247 + 99 = 346 pages

#4: If you write consecutive numbers starting with 1, what is the 50th digit you write? 

Solution I:
50 - 9 = 41, 9 being the first 9 digits you need to use for the first 9 pages.

Now it's 2 digit. 41/2 = 20.1 , which means you will be able to write 20 two digit numbers + the first digit of the next two digit numbers.

10 to 29 is the first 20 two digit numbers so the next digit 3 is the answer. (first digit of the two digit number 30.)

Solution II: (50 - 9 ) / 2 = 20. 5 ; 20.5 + 9 = 29.5, so 29 pages + the first digit of the next two digit numbers, which is 3, the answer.


#5: What is the sum if you add up all the digits from 1 to 100 inclusive?

00  10  20  30  40  50  60  70  80  90

01  11  21  31  41  51  61  71  81  91

02  12  22  32  42  52  62  72  82  92

03  13  23  33  43  53  63  73  83  93

04  14  24  34  44  54  64  74  84  94

05  15  25  35  45  55  65  75  85  95

06  16  26  36  46  56  66  76  86  96

07  17  27  37  47  57  67  77  87  97

08  18  28  38  48  58  68  78  88  98

09  19  29  39  49  59  69  79  89  99

Solution I:
Do you see the pattern?  From 00 to 99 if you just look at the unit digits.
There are 10 sets of ( 1+ 2 + 3 ... + 9) , which gives you the sum of 10 * 45 = 450
How about the tenth digits? There are another 10 sets of (1 + 2 + 3 + ...9) so another 450
Add them up and you have 450 * 2 = 900 digits from 1 to 99 inclusive.
900 + 1 ( for the "1" in the extra number 100) = 901 

Solution II:
If you add the digits on each column, you have an arithmetic sequence, which is
45 + 55 + 65 ... + 135  To find the sum, you use average * the terms (how many numbers)
\(\dfrac {45+135} {2} * \left( \dfrac {135-45} {10}+1\right)\) =900
900 + 1 = 901

Solution III :
2*45*10¹ + 1 = 901


Problems to practice: Answers below.

#1: A book has 213 pages, how many digits are there?

#2: A book has 1012 pages, how many digits are there?

#3: If you write down all the digits starting with 1 and in the end there are a: 100, b: 501 and c: 1196 digits, what is the last digit you write down for each question?

#4: What is the sum of all the digits counting from 1 to 123? 

Answers: 

#1: 531 digits. 

#2: 2941 digits.

#3: a. 5, b. 3, c. 3

#4: 1038 

Chicken, Rabbit Questions : Algebra Without Using Variables

The problems below are much easier so if you are preparing for the state, try the online timed test here.
Just write down random name to enter the test site is fine. At the end of the test, you'll see the answers for the ones you get wrong. Everyone is welcome to take the test. Thanks a lot and have fun problem solving. 

#1: There are 20 horses and chickens at Old Macdonald's farm. Together there are 58 legs. How many horses and how many chickens?

Solutions I :

#1: Using algebra, you have 

H + C = 20---equation 1 and

4H +2C = 58---equation 2

To get rid of one variable you can times equation 1 by 2 to get rid of Chicken or times 4 to get rid of horses. 

Times 2 and you have 2 H + 2 C = 40 (every term needs to be multiplied by 2)---equation 3

                                  4 H + 2C = 58---equation 2

Using equation 2 - equation 3 and yo get 2H = 18 so H = 9 and from there, solve for C = 11

Solution II: 

Let there be C chickens and (20-C) horses. [Since the sum of the number of chickens and horses is 20. If one has C number, the other has (20 - C)

2C + 4 (20 - C) = 58;  2C + 80 - 4C = 58; -2C = -22; C = 11 
and from there, you get H = 20 - 11 = 9

Solution III:
Without using algebra, you can make all the animals be chickens first. In that case, you'll have 20 x 2 = 40 legs.  Since you have 58 legs, you need to get rid of some chickens and bring in more horses. 

You gain 2 legs by every transaction (-2 + 4 = 2).  (58 - 40) / 2 = 9 so 9 horses and 11 chickens.

Other similar questions to practice (answer key below):  
 #1: Rabbits and ducks -- 30 animals and  86 feet. 

#2: There are 24 three-leg stools and four-leg tables. Together there are 86 legs.

#3: There are 43 bicycles and tricycles and together there are 100 wheels.

#4: There are 33 octopus (8 arms) and sea otters and together they have 188 arms/or for sea otters--legs. How many octopus and how many sea otters?  
#5: There are 18 animals in the barnyard, some are cows and some are chickens. There are total 48 legs. How many chickens and how many cows? 

 Answers: 

#1: 13 rabbits and 17 ducks. 

#2: 10 three-leg stools and 14 four-leg tables.

#3: 29 bicycles and 14 tricycles. 
#4: 14 octopus and 19 sea otters.  
#5:  6 cows and 12 chickens.

This Week's Work : Week 15 - for Inquisitive Young Mathletes

Part I work for this week:

See if you can write proof to show the exterior angle of any regular convex polygon is \(\frac{360}{n}\).
I'll include that in my blog for better proof.

Polygons Part I : interior angle, exterior angle, sum of all the interior angles in a polygon, how many diagonals
in a polygon

Polygons Part II : reviews and applicable word problems

Interior angles of polygons from "Math Is Fun"

Exterior angles of polygons from "Math Is Fun"

Supplementary angles

Complementary angles

Get an account from Alcumus and choose focus topics on "Polygon Angles" to practice.
Instant feedback is provided. This is by far the best place to learn problem solving, so make the best use of
these wonderful features.

This week's video on math or science : Moebius Transformations Revealed

Part II work online timed test word problems and link to key in the answers will be sent out through e-mail.

Have fun problem solving !!

This Week's Work : Week 8 -- for Inquisitive Young Mathletes

Assignment 1:
Using Algebra and Number Sense as Shortcuts from Mathcounts Mini

Watch the video and work on the activity sheet below the video on the same link for more practices.

 Also, review the following:
\(x^{2}-y^{2}=\left( x+y\right) \left( x-y\right)\) \(\left( x+y\right) ^{2}=x^{2}+2xy+y^{2}\) \(\left( x-y\right) ^{2}=x^{2}-2xy+y^{2}\) \(\left( x+y\right) ^{2}-2xy =x^{2}+ y^{2}\)
\(\left( x-y\right) ^{2}+ 2xy =x^{2}+y^{2}\)
\(\left( x+y\right) ^{2}-4xy =\left( x-y\right) ^{2}\)

Assignment 2:

Pascal's Triangle  from Math is Fun

Pascal's Triangle and Its Patterns

Assignment 3:
It'll be sent through e-mail.
Happy problem solving !!

This Week's Work : Week 4 -- for Inquisitive Young Mathleges

First of all, problem of the week from Evan, a 5th grader:
A man notices a sign in Shop-a-Lot that says: "All prices are marked 25% off today only!" He decides to buy a shirt that costs $65.12 before the discount. He then uses a $16 gift certificate and the clerk applies 12% sales tax. What is the final cost of the shirt after all the steps are applied? Express your answer to the nearest hundredth. 

This week, we'll learn two very common sequences : arithmetic and geometric sequences.
There are quite a few similarities between these two types and they are closely linked to ratio, proportion
so just watch the videos and play around/generate a few/ponder on those sequences. I don't expect you to learn them in just one week.

Notes from Regents Exam Prep: Arithmetic and Geometric Sequences and Series

From Mthcounts Mini:

Easier concepts:

Sequences

Arithmetic sequence/determine the nth term

Arithmetic and geometric sequences

Harder concepts:

Relationship between arithmetic sequences, mean and median

Sequences, series and patterns

From my blog : 
Some special arithmetic sequences and the easier way to find their sum

Write some notes of the most important features of the arithmetic sequence.
The best note will be posted here to share with other students.
Be creative !!

This Week's Work : week 3 -- for Inquisitive Young Mathletes

Assignment 1: 
Watch and learn Simon's Favorite Factoring Trick.

Work on some of the problems as well to check your understanding.

Assignment 2: 
Just learn as much as you can.
We'll keep practicing counting and probability.

Counting Permutations : from Art of Problem Solving

With or Without Replacement : from Art of Problem Solving

Notes on Permutations from the Math Page

Permutations with Some Identical Elements

This Week's Work : Week 2 -- for Inquisitve Young Mathletes

Assignment 1:
Mathcounts Mini related to the "Set" concept
Download the word problems below the video and work on them for this week.

Pascal's Triangle from Math is Fun.

Below is problem of the week, which continues with Evan's problem from last week so read it carefully.
Two players play a game starting with a pile of 26 sticks. The players alternate turns, each taking 1, 2, or 3 sticks on his or her turn.The player who takes the last stick wins.Who has the winning strategy in this game, the first player or the second player? How many sticks he/she needs to take? Why? 

Assignment 2:
Review special right triangles: Notes from my blog

30-60-90 Triangles from Art of Problem Solving

Powers of Pythagorean Triples from Art of Problem Solving

Working together rate  problems from Art of Problem Solving.

Mathcounts strategies: Some sums

The following sequences are all arithmetic sequences and for any arithmetic sequences, the sum is 

always average times the terms (how many numbers). 

To find the average, you can use 

a. sum divided by how many numbers.
b. average of the first and the last term, the second first and the second last term, or the third first and the third last term, etc...

Sum of the first consecutive natural numbers:

1 + 2 + 3 + 4 + 5 + ... + ( n -1 ) + n = \( \dfrac {n\left( n+1\right) } {2}\) 

Examples : 

#1: 1 + 2 + 3 + ... + 100 = \(\dfrac {100\left ( 101\right) } {2}=5050\)

#2: 1 + 2 + 3 + ... + 27 = \( \dfrac {27\left( 28\right) } {2}=378\)

#3: 4 + 8 + 12 + ... + 80 = 4 (1 + 2 + 3 + ... + 20) = \(\dfrac {4\times 20\times 21} {2}=840\)

Sum of the first consecutive natural even numbers: Proof without words 

Have you noticed in this sequence, every number is double the numbers in the first example,

so you don't need to divide by 2.  

\(2+4+6+\ldots +2n=n\left( n+1\right) \)

Examples: 

#1: 2 + 4 + 6 + ... 100 = 50 * 51 = 2550

#2: 2 + 4 + 6 + 8 + 10 + 12 = 6 x 7 = 42 

In this case, you can also find the midpoint, which is 7 and then 7 x 6 = 42

#3: 2 + 4 + 6 + ...420 = 210 * 211 = 44,310

Sum of the first consecutive odd numbers: Proof without words

In this special case, the mean is the same as how many numbers.

So it's easier if you find the mean by averaging the first and the last term and then square the mean.

1 + 3 + 5 + ... + ( 2n -1) = \(n^{2}\)

Examples: 

#1: 1 + 3 + 5 + ... + 39 = \(\left[ \dfrac {\left( 39+1\right) } {2}\right] ^{2}\) = (20^{2}\)

#2: 1 + 3 + 5 + ... + 89 = \(\left[ \dfrac {89+1} {2}\right] ^{2}=45^{2}=2025\)

#3: 1 + 3 + 5 + ... + 221 = \(\left[ \dfrac {221+1} {2}\right] ^{2}= 111^{2}=12321 \)

Applicable problems:

#1: What is the sum of the first 40 consecutive positive integers? 

#2: What is the sum of the first 40 consecutive positive even integers?

#3: What is the sum of the first 40 consecutive positive odd integers? 

#4: How many times does a 12-hour clock strikes in one day if it strikes once on one o'clock, twice on two o'clock, etc...?

#5: At a game show,you win $100 for the first correct answer and $200 for the second correct answer, etc. How much do you win if you answer 8 questions in a row correctly? 

Answer key:

#1: 820

#2: 1640

#3: 1600

#4: 156

#5: 100 + 200 + 300 ... + 800 = 100 (1 + 2 + 3 + ...8) = 100* \(\dfrac {8\times 9} {2}\) =  3600