Math Problem of the Month

[Kid J]

February 2009: Jack has fewer than 40 marbles in a bag. If he splits up the marbles among his 4 friends, he'll have 3 left over. If he divides them among his 7 friends, he'll have 2 left over. How many marbles are in the bag? (AMP-Countryside)

---

January 2009: What is the average of 1/8 and 1/10?

December 2008:
Mike can paint a room in 3 hours, and his son can paint the room in 6 hours. Working together, how long would it take them to paint the room?

November 2008:
How many 3 digit numbers have different digits in descending order (such as 732 or 960)?

October 2008:
A palindromic number reads the same forwards or backwards, such as 23532. What is the only palindromic prime number with an even number of digits?

September 2008:
A bell chimes the same number of chimes as the hour. If 6 chimes take 5 seconds, how long will it take to chime 12 times?

August 2008:
In how many ways can 7 identical marbles be placed in a red bag, a white bag, and a blue bag if there must be at least one marble in each bag?

July 2008:
What is the smallest positive integer greater than 1 which leaves a remainder of 1 when divided by 2,3,4,5,6 or 7?

[Kid J]

October, 2008:
A palindromic number reads the same forwards or backwards, such as 23532. What is the only palindromic prime number with an even number of digits?

ANSWER: 11
Solution:

All palindromic numbers with an even number of digits are divisible by 11. (Try it!) Therefore, the only prime palindrome with an even number of digits is 11 itself.

We could consider numbers with 2 digits (an even number of digits): The first two-digit prime is 11, which is also a palindrome and the answer to the question. Had that NOT worked, we could consider 33, 55, 77, and 99 (all composite). [Why not even consider 22, 44, 66, and 88?] Fortunately, we don't have to go on to the 4 digit numbers, 6 digit numbers, and beyond.

[Kid J]

September 2008:
A bell chimes the same number of chimes as the hour. If 6 chimes take 5 seconds, how long will it take to chime 12 times?

ANSWER: 11 seconds

Solution:

Think of a post and rail fence: If there are 6 posts, then there will be 5 rails between them. Similarly the bell chimes (fence post) and then waits (rails) to chime again. If 6 chimes take 5 seconds, there is 1 second between chimes. 12 chimes are separated by 11 waits, therefore, 12 chimes take 11 seconds.

[Kid J]

November 2008: How many 3 digit numbers have different digits in descending order (such as 732 or 960)?

ANSWER: 120

Solution:

Let's start small and work our way up.

The smallest hundred's digit possible is 2 (i.e. the number 210). The ten's digit could only be 1 and the only possibility for the unit's digit is 0. = 1 number.

Next the hundred's digit could be 3. Then the ten's digit could be 2, leaving 1 or 0 for the units = 2 numbers. Or the ten's digit could be 1, leaving only 0 for the unit's = 1. So the number of numbers beginning with 3 is 2+1 = 3.

For a hundred's digit of 4: The ten's digit could be 3, leaving the 3 possible unit's digits 2,1,and 0 = 3. Or the ten's digit could be 2, leaving the 2 possible unit's digits 1 and 0 = 2. Or, finally, the ten's digit could be 1, leaving only 0 for the unit's digit = 1. The number of numbers beginning with 4 is 3 + 2 + 1 = 6.

Consider the table form of this:

hundred's digit	number of 3-digit numbers	cumulative total
2	1= 1	1
3	2+1=3	4
4	3+2+1=6	10
5	4+3+2+1=10	20
6	5+4+3+2+1=15	35
7	6+5+4+3+2+1=21	56
8	7+6+5+4+3+2+1=28	84
9	8+7+6+5+4+3+2+1=36	120

[Kid A]

December 2008: Mike can paint a room in 3 hours, and his son can paint the room in 6 hours. Working together, how long would it take them to paint the room?

ANSWER: 2 hours

Solution 1: Use the Least Common Multiple

The LCM of 3 and 6 is 6. For convenience (i.e. miminal fraction use) we'll consider what Mike & his son can do in 6 hours. Mike can paint 2 rooms and his son can paint 1 more = 3 rooms painted in 6 hours. If we divide the 3 rooms by 3 to get the one room we want painted, we'll have to divide the 6 hours by 3 as well. This gives the answer of 2 hours needed to paint one room.

Solution 2: Traditional "Work Problem" Method - Rate of Work

Mike needs 3 hours to paint one room, so in 1 hour he paints 1/3 of a room. His son needs 6, so his hourly rate is 1/6 of a room per hour. Together 1/3 + 1/6 is 3/6 or 1/2, so together they paint 1/2 of a room per hour. They would need 2 hours to paint 1 whole room.

If this problem is too easy for you, please try to use the equation method to calculate the "together hours" when they work in a large room: father alone can paint the room is 7 hours, and his son alone can do it in 11 hours. How long (T) if they work together?
7xS1=1 (one big room)
11xS2=1 (the same big room)
or 7xS1=11xS2, therefore, S2=(7/11)xS1
When (S1+S2)xT=7xS1, then replacing S2 with the equation above:
(S1+(7/11)xS1))xT=7xS1, so
S1x(1+7/11)xT=7xS1, take the S1 from both site, so we get:
(18/11)xT=7, then move the numbers to the other side (by dividing (18/11) at both sides:
T=7/(18/11) or T=7*11/18=77/18=4.278
So the answer to a larger room is 4.278 hours.

[Kid J]

January 2009: What is the average of 1/8 and 1/10?

ANSWER: 9/80

Fight the temptation to jump to 1/9!!

Solution 1:
To average two numbers: add & divide by 2. 1/8 + 1/10 = 10/80 + 8/80 = 18/80.
Dividing by 2 is the same as multiplying by 1/2. 18/80 x 1/2 = 9/80

Solution2:
To more easily compare the two numbers, change to equivalent fractions with like demoninators. If you use 80, 1/8 = 10/80 while 1/10 = 8/80. Midway between 8/80 and 10/80 is the desired 9/80.

(Note: If you used the LEAST common denominator 40, you'd be finding the number midway between 4/40 and 5/40, which is 4.5/40. This equals 9/80)

Why ISN'T 1/9 the average?
Notice that the tempting answer 1/9 {which equals 9/81} is very close to 9/80. If you graph y = 1/x {some (x,y) points: (1,1/1), (2,1/2), (3,1/3), (4, 1/4), ...} you get part of an hyperbola. This is NOT a straight line. Halfway between (8, 1/8) and (9,1/9) {i.e. the midpoint of this segment} will be the average and will be just slightly under (8.5, 1/9).

[Kid J]

February 2009:
Jack has fewer than 40 marbles in a bag. If he splits up the marbles among his 4 friends, he'll have 3 left over. If he divides them among his 7 friends, he'll have 2 left over. How many marbles are in the bag? (AMP-Countryside)

ANSWER: 23

Solution:
List all the whole numbers (within 4) that can be divided by 4 (+3) and by 7 (+2):
By 4(+3): 4(7), 8(11), 12(15), 16(19), 20(23), 24(27), 28(31), 32(35), 36(39)
By 7(+2): 7(9), 14(16), 21(23), 28(30), 35(37)

You may do the listing and adding separately (or write the results in different lines).
You will see the two lines have only one number (23) in common.

[Kid A]

March 2009:

Kate drives to a distant city averaging 30 miles per hour, and returns over the same route at 40 miles per hour. What was his average rate for the entire trip? (MOEMS).

ANWSER: 35 mph.