Algebra orinQ; Opu'"'tion�with r2dic.;il(1) Given that x^3 - x = 0, factoring gives
x(x2 — 1 ) = x(x — l)(x + 1) = 0. Hence, x =
0 , x = 1, or x = -1. Since x > 0, the value
of x cannot be O or -1, and so x = 1;
SUFFICIENT.
(2) Given that嘉- x = 0, it follows that
嘉= x, or (嘉）^3 气，or x = x^3. Therefore,
x^3 - x = 0 and the discussion in (1) shows
that the only positive value of xis x = 1;
SUFFICIENT.
1h e correct answer 1s D·
each statement alone is sufficient.
0S08307- A total of 20 amounts are entered on a spreadsheet
that has 5 rows and 4 columns; each of the
2 0 positions in the spreadsheet contains one amount.
The average (arithmetic mean) of the amounts in row 1
is Ri (1 $ i $ 5). The average of the amounts in column
j is Ci (1 $ j $ 4). What is the average of all 20
amounts on the spreadsheet?
(1) R1 + R2 + R3 + R4 + R5 = 550
(2) C1 + C2 + C3 + C4 = 440Arithmetic'>l
It is given that Ri represents the average of the
amounts in row i. Since there are four amounts in
each row, 4Ri represents the total of the amounts
in row i. Likewise, it is given that 0 represents
the average of the amounts in column}. Since
there are five amounts in each column, 5 0
represents the total of the amounts in column}.(1) It is given that R 1 + R尸凡＋凡+Rs=
550, and so 4(R 1 + R 2 + R卢凡+Rs)=
4R 1 + 4R 2 + 4R 3 + 4R 4 + 4Rs = 4(550) =
2,200. Therefore, 2,200 is the sum of all
20 amounts (4 amounts in each of 5 rows),
and the average of all 20 amounts is
2,200 = 110·SUFFICIENT.
20 ，
(2) It is given that C 1 + C 2 + C 3 + C 4 = 440,
and so 5 (C 1 + C 2 + C 3 + C4) =
5C 1 + 5C 2 + 5C 3 + SC 4 = 5(440) = 2,200.
Therefore, 2,200 is the sum of all
20 amounts (5 amounts in each of5.5 ·.,':iutf· 1c1"1 Answer Explanations4 columns), and the average of all20 amounts 1s 2, 200 = 110; SUFFICIENT.
20
1h e correct answer 1s D·，
each statement alone is sufficient.
DS13132
230.Was the range of the amounts of money that
Company Y budgeted for its projects last year equal to
the range of the amounts of money that it budgeted for
its projects this year?(1) Both last year and this year, Company Y
budgeted money for 12 projects and the least
amount of money that it budgeted for a project
was $400.
(2) Both last year and this year, the average
(arithmetic mean) amount of money that
Company Y budgeted per project was $2,000.Arithmetic ir•:--
Let G 1 and L 1 represent the greatest and least
amounts, respectively, of money that Company Y
budgeted for its projects last year, and let G 2
and L 2 represent the greatest and least amounts,
respectively, of money that Company Y budgeted
for its projects this year. Determine if the range
of the amounts of money Company Y budgeted
for its projects last year is equal to the range of
amounts budgeted for its projects this year; that
is, determine if G 1 - L 1 = G 2 - L 2.(1) This indicates that L 1 = L 2 = $400, but does
not give any information about G 1 or G忒
NOT sufficient.
(2) This indicates that the average amount
Company Y budgeted for its projects both
last year and this year was $2,000 per
project, but does not give any information
about the least and greatest amounts that it
budgeted for its projects either year; NOT
sufficient.Taking (1) and ( (^2) ) together, it is known that
L 1 = L 2 = $400 and that the average amount
Company Y budgeted for its projects both last
year and this year was $2,000 per project, but
there is no information about G 1 or G 2. For
example, if, for each year, Company Y budgeted
$400 for each of 2 projects and $2,320 for each
of the 10 others, then (1) and (2) are true and the