Difference between revisions of "2002 AMC 12B Problems/Problem 2"

Latest revision as of 09:46, 9 August 2022

The following problem is from both the 2002 AMC 12B #2 and 2002 AMC 10B #4, so both problems redirect to this page.

What is the value of $(3x - 2)(4x + 1) - (3x - 2)4x + 1$ when $x=4$ ?

$\mathrm{(A)}\ 0 \qquad\mathrm{(B)}\ 1 \qquad\mathrm{(C)}\ 10 \qquad\mathrm{(D)}\ 11 \qquad\mathrm{(E)}\ 12$

By the distributive property,

$(3x-2)[(4x+1)-4x] + 1 = 3x-2 + 1 = 3x-1 = 3(4) - 1 = \boxed{\mathrm{(D)}\ 11}$ .

Inputting 4 into $x$ in the original equation,

$[3(4)-2][4(4)+1]-[3(4)-2][4(4)]+1 = 170-160+1 = \boxed{\mathrm{(D)}\ 11}$

2002 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 3	Followed by Problem 5
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

2002 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 1	Followed by Problem 3
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

@@ Line 9: / Line 9: @@
 \qquad\mathrm{(D)}\ 11
 \qquad\mathrm{(E)}\ 12</math>
-== Solution ==
+== Solution 1 ==
 By the distributive property,
-<cmath>(3x-2)[(4x+1)-4x] + 1 = 3x-2 + 1 = 3x-1 = 3(4) - 1 = \boxed{\mathrm{(D)}\ 11}</cmath>
+<cmath>(3x-2)[(4x+1)-4x] + 1 = 3x-2 + 1 = 3x-1 = 3(4) - 1 = \boxed{\mathrm{(D)}\ 11}</cmath>.
-==Comment==
+== Solution 2 ==
-It would be nicer if the organizers chose a more difficult substitution, say <math>x=4.7</math>, to penalize those solutions that do not take advantage of the distributive property.
+Inputting 4 into <math>x</math> in the original equation,
+<cmath>[3(4)-2][4(4)+1]-[3(4)-2][4(4)]+1 = 170-160+1 = \boxed{\mathrm{(D)}\ 11}</cmath>
 == See also ==