The numbernsatisfies the equation√80 +√125 =√n. What is the value ofn?
(A)205(B)305(C)405(D)500(E)1000
2010 AMC 10 B, Problem #6
—
2010 AMC 12 B, Problem #4
—
“Note that
√
80 = 4
√
5
and
√
125 = 5
√
5
.”
Solution
Answer (C):
Simplifying gives
√
80 +
√
125 = 4
√
5 + 5
√
5 = 9
√
5 =
√
405
. Hence
n
= 405
.
Difficulty:
Medium-easy
NCTM Standard:
Number and Operations Standard for Grades 9–12: understand numbers, ways
of representing numbers, relationships among numbers, and number systems.
The square ofxis 16 larger than the square ofy, andxis 2 larger thany. What is the productxy?
2010 AMC 10 B, Problem #7—“Note thatx2−y2= (x−y)(x+y) = 16.”
Difficulty:
Medium-easy
NCTM
Standard:
Algebra Standard for Grades 9–12:
write equivalent forms of equations,
inequalities, and systems of equations and solve them with fluency-mentally or with paper and
pencil in simple cases and using technology in all cases.
The area of triangleEBDis one third of the area of3–4–5triangleABC. SegmentDEis perpendicularto segmentAB. What isBD?ABCDE345(A)43(B)√5(C)94(D)4√33(E)522010 AMC 10 B, Problem #8—“The area of△ABCis12·3·4 = 6.”
Standard:
Geometry Standard for Grades 9–12:
analyze properties and determine
attributes of two- and three-dimensional objects.
If a rectangle’s width is increased by 3, the rectanglebecomes a square and the area of the rectangle isincreased by 54.What is the area of the originalrectangle?
(A)90(B)150(C)180(D)225(E)270
2010 AMC 10 B, Problem #9—“Letxandybe the width and length of the originalrectangle, respectively. Form an equation to show thedifference in areas.”
Standard:
Geometry Standard for Grades 9–12:
analyze properties and determine
attributes of two- and three-dimensional objects.
