**Lower Elementary**

*Question: *Ben walked 2 miles to school then half a mile to his friend’s house. He then walked back to school where he was picked up by his mom and taken home. How far did Ben walk?

*Answer: *3 miles

*Solution: *A whole is the sum of its parts. We can identify the whole as the total distance walked and the parts as the distance walked to school, the distance walked to his friend’s house, and then the distance walked back to school from his friend’s house.

Total distance walked = distance walked to school + the distance walked to friend’s house + distance from friend’s house back to school

Total distance walked = 2 miles + ½ mile + ½ mile = 3 miles

**Upper Elementary**

*Question: *Benny owns a third as many baseball cards as Pat. Pat owns a quarter as many baseball cards as Sammy. Sammy owns half as many baseball cards as Jimmy. If Jimmy owns 120 baseball cards how many cards do Benny, Pat, and Sammy own?

*Answer: *Benny: 5, Pat: 15, Sammy: 60

*Solution: *Since Jimmy is the only one with a known amount of cards, we will start with him and work backwards. Sammy has half as many cards as Jimmy. Taking a half of Jimmy’s 120 cards, we see that Sammy has 60 cards. Pat has a quarter of the amount of cards Sammy has. Since a quarter is a half of a half, we take half of a half of Sammy’s 60 cards to find that Pat has only 15 cards. Lastly Benny has a third of the amount of cards Pat has. Breaking 15 up into three equal groups, one group has 5 cards so Benny has 5 cards.

**Middle School**

*Question: *A salesman sold $10,392 worth of electronics equipment last month. If he makes 4% commission on all of his sales and earns a monthly salary of $1,600, what was his total pay last month?

*Answer: *$2015.68

*Solution: *To find how much commission the salesman makes, we need to find 4% of $10,392.

$10,392(.04) = $415.68

Because he makes a monthly salary of $1,600, we add his commission last month to his salary to find his total pay.

$1,600 + $415.68 = $2015.68

**Algebra and Up**

*Question: *Find h.

*Answer: *h = 24

*Solution: *In order to solve for h we need to break the base up into 2 separate bases with unknown lengths.

The larger triangle has a base of 42 – x while the smaller triangle has a base of just x. Using the Pythagorean Theorem we can write two equations with two unknowns and then solve the system simultaneously.

a^{2} + b^{2} = c^{2}

{ (42 – x)^{2} + h^{2} = 40^{2}

{

{ x^{2} + h^{2} = 26^{2}

Expanding out the first equation:

1764 – 84x + x^{2} + h^{2} = 40^{2}

There are various ways of solving this system of equations (ie. substitution, elimination, matrices etc.), but in this example we will use the substitution method. Using the second equation we can solve for x^{2} and x.

x^{2}2 = 26^{2} – h^{2}

x = √(26^{2} – h^{2})

Plugging these into the expanded first equation we now have 1 equation written with one unknown.

1764 – 84√(26^{2} – h^{2}) + 26^{2} – h^{2} + h^{2} = 40^{2}

Simplify.

1764 – 84√(676 – h^{2}) + 676= 1600

-84√(676 – h^{2}) = -840

√(676 – h^{2}) = -840/-84 = 10

676 – h^{2} = 100

-h^{2} = -576

h^{2} = 576

h = √576 = 24