**Lower Elementary:**

*Question: *A ball pit starts the day with 140 balls. At the end of the day, 50 balls fell out of the pit. The workers were able to put 28 of the balls back in the pit. How many balls are in the pit now?

*Answer: *118 balls

*Solution: *First we need to find how many balls were left before the workers put the balls back. We subtract the balls that fell out by the starting number of balls. 140 – 50 = 90. There were 90 balls left in the pit before the workers put in some of the balls. They put in 28 balls, so 90 + 28 = 118. There were 118 balls in the pit after the workers put in the balls.

**Upper Elementary:**

*Question: *Becky used different colored beads to express numbers. The blue beads represent the ten thousands place, the red beads represent the thousands place, the yellow beads represent the hundreds place, the purple beads represent the tens place, and the green beads represent the ones place. What number is represented by 6 blue beads, 8 red beads, 12 yellow beads, 5 purple beads, and 7 green beads?

*Answer: *69,257

*Solution: *Each of the beads represents a place value. Using the beads and their corresponding place values, we have 6 ten thousands, 8 thousands, 12 hundreds, 5 tens and 7 ones. Notice that 10 hundreds = 1 thousand. 12 hundreds = 1 thousand and 2 hundreds, so that means our number is 6 ten thousands, 9 thousands, 2 hundreds, 5 tens and 7 ones, or 69,257.

**Middle School:**

*Question: *If 4/9 of a gallon of water fills a bottle 3/5 full, how many gallons of water are needed to fill the bottle?

*Answer: *20/27 of a gallon

*Solution: *One way to solve this problem is to find how many gallons fill one fifth of the bottle and then multiply it by 5 so that we can have 5/5, or the whole, bottle. 4/9 of a gallon fills 3/5 of the bottle. We divide 4/9 by 3 to see how many gallons fill 1/5 of the bottle.

4/9 ÷ 3

Dividing a fraction by 3 is the same as multiplying by 1/3

4/9 × 1/3 = 4/27

So 4/27 of a gallon fills 1/5 of the bottle. To find out how much fills 5/5 of the bottle, or the whole bottle, multiply by 5.

4/27 × 5 = 20/27

So 20/27 of a gallon fills the whole bottle.

**Algebra and Up:**

*Question: *Solve for x: 4^{3x + 2} = 8^{4x – 3}

*Answer: *13/6 = 2 1/6

*Solution: *One way to solve this problem is to get both sides of the equation to have the same base. Notice that 4 = 2^{2} and 8 = 2^{3}. When an exponent is raised to another exponent, we multiply the exponents. So, now we have:

2^{2(3x + 2)} = 2^{3(4x – 3)}

Since the bases are the same, the exponents must be the same.

2(3x + 2) = 3(4x – 3)

Distribute.

6x + 4 = 12x – 9

Move the x to one side and the other numbers to the other side.

13 = 6x

Divide by 6.

2 1/6 = x

So x = 2 1/6 = 13/6.