Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2017 Grade 5 Number System
Problem
Solution
1) Malou has a collection action figures. If she puts them in groups of 2, there is 1 left over. If she puts them in groups of 3, there is 1 left over. When she puts them in groups of 5, there is 1 left over again. What is the least number of figures she can have? Find the least common multiple (LCM) of 2,3 and 5 and then add 1 to it.
The LCM of 2,3 and 5 = 30 + 1 = 31

2) Collectible cards come in packs of 4 or packs of 9. Vidan needs to buy exactly 110 cards. He wants to buy as many of the packs of 9 as he can as they are a better buy. How many packs of 9 should he buy?		 1. Start with 110/9 =12 packs of 9 cards and compute the remainder = 2.
2. Do the same calculation, decreasing the number of 9-card packs until your remainder is a multiple of 4. The number of 9-packs is 110/(9 X 10) = 20, a multiple of 4, so the number of 9-card packs is 10
3) Hunter, Brock, Sparrow, and Dane all brought pencils to share. Brock brought 3 more than Hunter. Sparrow brought 4 more than Brock, and Dane brought 3 more than Sparrow. They share the pencils evenly between the four of them. How many pencils will Hunter gain? 1. Assume that Hunter brought 1 pencil.
2. Then Brock brought 4 pencils.
3. Sparrow brought 8 pencils, and
4. Dane brought 11 pencils.
5. Add up these number of pencils and divide by 4 =
    1 + 4 + 8 + 11 = 24/4 = 6 pencils each.
6. Subtract 1 and Hunter's gain was 5 pencils.

Problem
Solution
4) Rigoberto has a large rectangular board with sides 25 inches by 35 inches. He is arranging smaller squares on it in a grid. The side of each smaller square is 2 3/4 inches long. He wants to fit as many of the smaller squares onto the board as he can, but he can neither cut the smaller squares, nor let them overlap. How many of the smaller squares can he fit on the large board? 1. Divide 25 by 2 3/4 and round down = 9.
2. Divide 35 by 2 3/4 and round down = 12.
3. Multiply these together to get the total number of squares that will fit in the rectangle = 108.

5) Jude and her friends mowed lawns and were paid with a bag of quarters. They give 3 quarters to Jude's little brother. Anouk then takes 1/5 of the remaining quarters. Then Sara takes 1/4 of what is left. Jude gets the rest, which is 1/2 of the original number of quarters. How many quarters did they originally get?		 Let N = the original number of quarters in the bag.

1. Then N - 3 are left after Jude gave 3 to his little brother.
2. Anouk takes 1/5 of the remainder leaving 4/5 of (N - 3)
3. Sara takes 1/4 of what's left leaving 3/4 of what Anouk left =
    3/4 X 4/5 X (N - 3) = 3/5(N - 3).
4. This is N/2 which is what Jude got
5. Take what Sara left and make it equal to what Jude got and solve for N:
    3/5 (N - 3) = N / 2
    3(N - 3) = 5N/2
    3N - 9 = 5N/2
    6N - 18 = 5N
    N = 18