Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2012 Grade 7 Number Sense
Problem
Solution
1) Jimmy is 10/9 as tall as Ying'ai. Ishmael is 4/5 as tall as Amanda. The difference between Amanda and Ying'ai is 4 inches. Ishmael is 14 inches shorter than Ying'ai. Ying'ai is 4 feet 6 inches tall. How tall is Jimmy? With this type of problem you start at the bottom and work backwards.
  1. Ying'ai is 4 feet 6 inches = 54 inches.
  2. Since all you want is Jimmy's height, you can forget Ishmael and Amanda.
  3. Jimmy is 10/9 x 54 = 60 inches = 5 feet tall

2) A two digit number is divided by a two digit number with quotient a one digit number. The five digits used are 1, 2, 4, 7, 8 and each is used only once. What is the larger two digit number?
  1. The larger number must start with a 4, 7 or 8 for it to be divided by another number to make a single digit quotient.
  2. It must also be even and end in 2,4 or 8 for it to be divisible evenly by the remaining numbers because there is no 3 here.
  3. If it starts with a 4 then it must be 42 or 48, neither of which is divisible by the remaining 3 digits: 1,7 or 8, so the number must start with a 7 or 8
  4. List the possibilities:
        72 remaining digits 1,4,8
        72/18 = 4! This works!
        78 remaining digits 1,4 7: doesn't work
  5. The higher number is 72
3) Donna has 1/8 tank of gas remaining when she pulls into a gas station. After she puts 15 gallons of gas in her car, the gas gauge reads 3/4 full. How many gallons does Donna's tank hold?
  1. The 15 gallons occupied
    3/4 - 1/8 = 6/8 - 1/8 = 5/8 of the tank.
  2. (5/8) T = 15 gallons; T = 15x8/5 = 24 gallons

Problem
Solution
4) One tactic for adding fractions with complicated denominators is to use their prime factorizations to find a least common multiple. Try that here: What is the denominator for the sum
    a     +     b  
  924       364
when expressed as a single fraction in lowest terms?
  1. Prime factorization for 924:
    924 = 462x2 = 231x2x2 = 33x7x2x2 = 11x3x7x2x2
  2. Prime factorization for 364:
    364 = 182x2 = 91x2x2 = 13x7x2x2
  3. Common prime factors = 7x2x2 = 28
  4. So the greatest common factor (GCF) of 924 and 364 is 28
  5. Using the GCF-LCM equation:
    LCM (924,364) = 924x364 / 28 = 12012
5) What is the smallest whole number n so that
  1. Compute (-3⁄4)n for n = 1,2,3, ... until it's absolute value falls below 1⁄3:
  2. (-3⁄4) 2 = +.5625
  3. (-3⁄4) 3 = -.421875
  4. (-3⁄4) 4 = +.31640 <=== below 1⁄3
  5. n = 4