Competition · AMC preparation · step 4 of 4
AMC 8 · 2008 · #7
Grade 4 arithmeticrate-ratioPick an answer.
AMC 8 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The chain 3/5 = M/45 = 60/N is really two independent statements glued together. Tool #7 (Identify Subproblems) splits it into one equation for M and one equation for N. Each piece is a Grade 4 equivalent-fraction question: scale the top and bottom of 3/5 by the same factor. Tool #3 (Write an Equation) lets us name that factor in each case and solve quickly.
Split into two equations
Split the chain into two separate equations, one for M and one for N.
The Grade 4 equivalent-fraction rule says two fractions are equal when one comes from the other by multiplying top and bottom by the same number. Each sub-equation can be solved on its own.
4.NF.A.1Identify SubproblemsSolve for M
The denominator 5 → 45 is ×9, so scale the numerator by the same 9: M = 27.
Asking ``5 times what is 45?'' is Grade 3 unknown-factor work. The same multiplier scales the top.
Because the denominator 5 is multiplied by 9 to reach 45, the numerator 3 must be multiplied by the same 9, so M = 27.
▸ Why?
Two fractions are equal only when one is the other with the top and the bottom multiplied by the same number, so whatever multiplies 5 up to 45 has to multiply the 3 on top as well.
▸ Why?
Multiplying the top and the bottom by the same 9 leaves the fraction's value unchanged: cutting each part into 9 makes both counts nine times larger while the portion of the whole stays the same.
▸ Why?
Cutting each of the 5 parts, and each of the 3 shaded parts, into 9 equal pieces makes each count nine times as big — 5 becomes 45 and 3 becomes 27 — since counting equal groups is multiplication.
▸ Why?
The finer pieces still fill the very same whole with no gaps and no overlaps, so the shaded region is the same size as before — only the number of pieces it is cut into has changed.
▸ Why?
The shared multiplier is 9 because that is the number which, times 5, gives 45; we recover it by dividing 45 by 5, the operation that undoes the multiplication.
Solve for N
The numerator 3 → 60 is ×20, so scale the denominator by the same 20: N = 100.
Same idea, with the unknown on the bottom this time. The multiplier is whatever turns 3 into 60.
4.NF.A.1Eliminate PossibilitiesAdd M and N
Add the two results: M + N = 27 + 100 = 127.
Standard Grade 4 multi-digit addition.
4.NBT.B.4Eliminate PossibilitiesWhen two fractions are equal, the top and bottom always scale by the same number — find that scale factor and the unknown falls out.
- Split into two equations
- Solve for M
- Solve for N
- Add M and N
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