**Lower Elementary:**

*Question: *While walking down the street, Nancy noticed a pattern in the tiles in the sidewalk. The tiles she walked by were the following colors: green, blue, green, red, green, blue, green, red, green. If the pattern continues, what are the colors of the next 2 tiles?

*Answer: *blue and then green

*Solution: *The pattern of the tiles is green, blue, green, red, and then it repeats. The last tile that Nancy walked by was the 1st green, so the next 2 tiles would be blue and then green.

**Upper Elementary:**

*Question: *Dale planted 4 banana trees. Each tree gave 15 bananas. He sold each banana for $1.25. If he sold all the bananas, how much money did Dale make?

*Answer: *$75

*Solution: *To find the total amount of bananas that Dale has, multiply the number of trees times the number of bananas per tree. 15 × 4 = 60 bananas. To get the total amount that he makes selling all the bananas, multiply the price of each banana times the total number of bananas. $1.25 × 60 bananas = $75.

**Middle School:**

*Question: *Jason, Gary, and Alyssa worked together to buy speakers for their band. Jason put in $50, Gary put in $75, and Alyssa put in $25. They decided to then sell the speakers for $120. How much of the selling price should each person receive so that it is fair?

*Answer: *Jason: $40, Gary: $60, Alyssa: $20

*Solution: *To make it fair, each person should receive the fractional part that they put in to buy the speakers. The total amount of the speakers is $150 ($50 + $75 + $25). Jason paid for 1/3 of the speakers (50/150 = 1/3), Gary paid for 1/2 of the speakers (75/150 = 1/2), and Alyssa paid for 1/6 of the speakers (25/150 = 1/6). To find how much each person should receive, find each of the respective fractional parts of $120.

Jason: 1/3 of $120 = $40

Gary: 1/2 of $120 = $60

Alyssa: 1/6 of $120 = $20

**Algebra and Up:**

*Question: *Paula has a 30% concentrate apple juice and a 45% concentrate of apple juice. How much of each does she need to mix together to have a 2 liter amount that is 40% concentrate apple juice?

*Answer: *2/3 of a liter of the 30% concentrate and 1 1/3 of a liter of the 45% concentrate

*Solution: *One way to solve this problem is to set up 2 equations. Let x be the amount (in liters) of the 30% concentrate and let y be the amount (in liters) of the 45% concentrate. The two equations are:

1) The whole is equal to the sum of its parts: (amount of the 30% concentration) + (amount of the 45% concentration) = (amount of the 40% concentration): x + y = 2

2) Quantity and denomination: (concentration of 30%) × (amount of 30%) + (concentration of 45%) × (amount of 45%) = (concentration of 40%) × (amount of 40%): 0.3x + 0.45y = 0.4(2)

Let’s solve this system of equations using elimination (you may also solve this using other methods such as substitution or graphing):

x + y = 2

0.3x + 0.45y = 0.8

Multiply the top equation by –0.3:

–0.3x + –0.3y = –0.6

0.3x + 0.45y = 0.8

Add the two equations (so that the x can cancel):

0.15y = 0.2

Divide both sides by 0.15 to solve for y:

y = 4/3 = 1 1/3

Plug this into the equation (1) and solve for x:

x + 1 1/3 = 2

x = 2/3

Paula needs 2/3 of a liter of the 30% concentrate and 1 1/3 of a liter of the 45% concentrate to make a 2 liter amount of 40% concentrate.