AMC 10 · 2009 · #15

Grade 8 algebra

Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity

systems-of-equationslinear-equations-two-varfraction-arithmetic convert-to-algebra ↑ Prerequisites: systems-of-equations

📏 Medium solution 💡 2 insights

Problem

A bucket holds some fixed weight of its own, plus water. When it is two-thirds full the bucket and water together weigh a kilograms; when it is one-half full they weigh b kilograms. Find the total weight, in terms of a and b, when the bucket is completely full.

Pick an answer.

(A)

$\ \frac23a + \frac13b$

(B)

$\ \frac32a - \frac12b$

(C)

$\ \frac32a + b$

(D)

$\ \frac32a + 2b$

(E)

$\ 3a - 2b$

AMC 10 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Introduce a Variable

Two hidden quantities never change: the bucket's own weight and the weight of a full load of water. Name each with a variable, turn the two 'how full' facts into two equations, and the full-bucket weight falls out by combining them.

1STEP 1

Name the two hidden weights

Let B be the empty bucket's weight and w the water in a full bucket. Both stay fixed however full it is.

B = empty bucket, w = full load of water

2STEP 2

Write each fullness as an equation

Two-thirds full gives $B + \frac{2}{3}w = a$ ; one-half full gives $B + \frac{1}{2}w = b$ .

B + 2/3 w = a, B + 1/2 w = b

3STEP 3

Subtract to isolate the water

Subtracting kills B and leaves $\frac{1}{6}w = a - b$ , so a full load of water is 6a - 6b.

(2/3 - 1/2)w = a - b → 1/6 w = a - b → w = 6a - 6b

4STEP 4

Recover the bucket weight

Put that back into $B + \frac{1}{2}w = b$ : the empty bucket weighs 4b - 3a.

B = b - 1/2(6a - 6b) = 4b - 3a

5STEP 5

Add bucket plus full water

Full weight is $B + w = (4b - 3a) + (6a - 6b)$ , which is choice (E), 3a - 2b.

B + w = (4b - 3a) + (6a - 6b) = 3a - 2b

Answer

3a - 2b

Test with real numbers: let the empty bucket be 2 and a full load of water be 6. Then two-thirds full is 2 + 4 = 6 = a and one-half full is 2 + 3 = 5 = b. A full bucket is 2 + 6 = 8. The formula gives 3a - 2b = 18 - 10 = 8, which matches, so (E) is correct.

💡Key takeaway

Name the parts that never change, write one equation per clue, then subtract to make the part you don't want disappear.

Name the two hidden weights
Write each fullness as an equation
Subtract to isolate the water
Recover the bucket weight
Add bucket plus full water

Name the two hidden weights

Write each fullness as an equation

Subtract to isolate the water

Recover the bucket weight

Add bucket plus full water

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