AMC 10 · 2004 · #5
Grade 7 arithmeticPick an answer.
This is a word problem with one unknown d tangled inside a story about money, the signature trigger for Tool #13 (Convert to Algebra): turn the sentences into one equation and let the algebra do the work. Tool #8 (Analyze the Units) handles the exchange rate — tracking how 7 U.S. dollars turn into 10 Canadian dollars keeps the conversion honest. Tool #4 (Introduce a Variable) lets d carry the unknown from the setup straight through the solve. Finally the question has two layers, so Tool #7 (Identify Subproblems) splits it: first find d, then answer what is actually asked — the sum of its digits.
Turn the rate into a gain
The rate adds three sevenths to whatever she exchanges.
A fixed rate scales every dollar the same way, so the whole pile grows by the same fraction one small batch does.
6.RP.A.3Analyze The UnitsWrite the story as an equation
Landing back on the same number means that gain equals the 60 she spent.
If spending 60 brings her back to her starting number, then the exchange must have handed her exactly 60 extra.
7.EE.B.4Convert To AlgebraSolve for the amount
Solving that gives the starting amount as 140.
Undo a fraction of a number by splitting into equal parts: if three parts make 60, one part is 20 and seven parts is 140.
Because the exchange added a fixed fraction of her money, knowing that added amount hands back the whole.
▸ Why?
A fraction of a quantity is that quantity cut into equal shares, so knowing several shares tells you one share and then all of them.
▸ Why?
The same operation applied to both sides keeps an equation true, so scaling up from one share is safe.
Answer what was asked
The digits add to 5, choice (A).
Read the last small task literally — the digit sum, not the number — so the right work does not get thrown away on the wrong question.
2.NBT.A.1Identify SubproblemsA fixed exchange rate grows your money by a fixed fraction, so turn the rate into that fraction, set the extra equal to what you spent, then answer the exact question asked — here the digit sum, not the number.
- Turn the rate into a gain
- Write the story as an equation
- Solve for the amount
- Answer what was asked