1)  In the following sequence of numbers, each number has one more 1 than the preceding number: 1, 11, 111, 1111, 11111, ...  What is the tens digit of the sum of the first 30 numbers of the sequence?
2)  When asked how many gold coins he had, the collector said:
     If I arrange them in stacks of five, none are left over.
     If I arrange them in stacks of six, none are left over.
     If I arrange them in stacks of seven, one is left over.
     What is the least number of coins he could have?
3)  In the subtraction problem below, each letter represents a digit, and different letters represent different digits. What digit does C represent?
     A B A
   -    C A
        A B



Solutions

1)  The ones column of the 30 numbers contains 30 ones making a sum of 30. Thus, the ones digit of the sum is 0, carry 3. The tens column contains 29 ones. Its sum is 29 plus the 3 from the "carry," making 32. Therefore the tens digit of the sum is 2.
2)  The number of coins must be a multiple of 30. The multiples of 30 are 30, 60, 90, 120, 150, and so on. The smallest of these multiples that leaves a remainder of 1 when divided by 7 is 120.
3)  It is clear that B = 0 and A = 1. Substitute those numbers for letters as shown below. Therefore C is 9.
   1 0 1
-    C 1
      1 0

to edit.

Primary:
1 Dot: Can you make 7 using 14 and 7 or 3 and 9? How?
2 Dots: Can you make 5 using 11, 3, and 2 or 2, 5, and 8? How?
3 Dots: Can you make 24 using 8, 16, and 1 or 15, 3, and 6? How?

Intermediate:
1 Dot: How can you make 24 using 6, 5, 7, and 6?
2 Dots: How can you make 24 using 6, 8, 7, and 2?
3 Dots: How can you make 24 using 14, 3, 21, and 9?

Primary:
​1 Dot: Can you make 5 using 4 and 0 or 7 and 2? How? 
​2 Dots: Can you make 10 using 8/3/1 or 12/3/6? How?
​3 Dots: Can you make 24 using 8/16/1 or 15/3/6? How?

Intermediate:
1 Dot: How can you make 24 using 1, 2, 6, and 7?
​2 Dots: How can you make 24 using 6, 4, 1, and 4?
​3 Dots: How can you make 24 using 14, 3, 21, and 9?

1)  A girl bought a dog for $10, sold it for $15, bought it back for $20, and finally sold it for $25. Did the girl make or lose money, and how much did she make or lose?
2)  The average of five weights is 13 grams. This set of five weights is then increased by another weight of 7 grams. What is the average of the six weights?
 
3)  Each of the boxes in the figure is a square. How many different squares can be traced using the lines in the figure?


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Solutions
1)  She paid out $10 + $20 = $30. She received $15 + $25 = $40. She made $10.
2)  The average of the five weights is 13 grams. Then the total weight of the five weights is 5 x 13 or 65 grams. The sixth weight increases the total to 72 grams. The average of the six weights is 72/6 or 12 grams.
 
3)  There are three different sizes for the squares that can be traced in the figure: 1 x 1, 2 x 2, and 3 x 3. The table shows how many squares can be traced for each size. 

Primary:
​1 Dot: Can you make 9 using 12 and 4 or 18 and 9?
2 Dots: Can you make 6 using 9, 4, and 12 or 10, 8, and 12?
​3 Dots: Can you make 24 using 18, 2, and 4 or 5, 10, and 21?

​Intermediate: 
1 Dot: How can you make 24 using 1, 7, 2, and 2?
2 Dots: How can you make 24 using 7, 1, 3, and 3?
​3 Dots: How can you make 24 using 14, 3, 9, and 21? 

1)  Eight people want to play a 48-minute game as a team but only a team of exactly five is allowed to play. However, during the game, a player may be replaced by someone else. Suppose each of the eight people plays in the game for the same amount of time. How many minutes will each of the eight people play?
2)  A certain brand of sardines is usually sold at 3 cans for $2. Suppose the price is changed to 4 cans for $2.50. Will the new cost for 12 cans be more or less than the usual cost for 12 cans, and by how much?  
3)  A work crew of 3 people requires 3 weeks and 2 days to do a certain job. How long would it take a work crew of 4 people to do the same job if each person of both crews works at the same rate as each of the others? Note: each week contains six work days.


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Solutions
1)  Since the game-time is 48 minutes, the total playing time for the five active players is 5 x 48 = 240 minutes. If eight players share the total playing time, each player will play 240/8 = 30 minutes.
2)  The cost of 12 cans at the old rate was 4 x $2 or $8. The cost of 12 cans at the new rate was 3 x $2.50 or $7.50. The new price for 12 cans is $0.50 less than the old price for 12 cans.
3)  Each person of the work crew of three people worked 20 days. Thus the number of individual work days needed to do the job was 60. Then each member of the work crew of four people must work 15 days in order to provide a total of 60 individual work days.

Click on the Number Pieces button to access virtual base ten blocks that students can use to help them with solving problems. 

1)  A freight train travels 1 mile in 1 minute 30 seconds. At this rate, how many miles will the train travel in 1 hour?
2)  An Olympiad team is made up of students from the 4th, 5th, and 6th grades only. Seven students are 5th graders, eleven students are 6th graders, and one-third of the entire team are 4th graders. How many students are on the team?
3)  There is an even number between 200 and 300 that is divisible by 5 and also by 9. What is that number?




Solutions 
1)  Method 1: The train travels 1 mile in 1 minute 30 seconds. Then it will travel 2 miles in 3 minutes. Since 60 minutes contains 20 groups of 3 minutes, the train will travel 20 x 2 = 40 miles in 1 hour.
     Method 2: The number of miles the train travels in 1 hour is equal to 60 minutes divided by 1.5 minutes. 60 divided by 1.5 = 40.
2)  The total number of students in the 5th and 6th grades, 7 + 11 = 18, must be 2/3 of the team. Then 1/3 of the team is 9, and 3/3 or the entire team has 3 x 9 = 27 students.
3)  Since the number is divisible by 2, 5, and 9, it is also divisible by the product 2 x 5 x 9 = 90. The multiple of 90 between 200 and 300 is 270.