Determine the price of two type A footballs if the total cost of a type A and a type B football is $500.
1. Type B football costs $200.
2. Two type A and three type B footballs costs $1200.
Let A and B represent the number of type A and Type B footballs respectively. Then A + B = 500.
In statement 1, B = 200, hence, from A + B = 500, A = 500  B = 500  200 = $300. Thus 2A = $600. The statement is sufficient.
In statement 2, 2A + 3B = 1200, from A + B = 500,multiplying it by 3, we have 3A + 3B = 1500. Subtracting 2A + 3B = 1200 from 3A + 3B = 1500, we have A = 300; thus 2A = $600. The statement too is sufficient.
What is the value of the positive number, p?
1. One of its divisors is 7.
2. p is divisible by two positive numbers only
In statement 1, since one of the divisor is 7, it implies that the number is a multiple of 7. This allows us to have infinitely many numbers, hence the statement is not sufficient.
In statement 2, the number is divisible by two numbers only. This implies that p is prime since a prime number is divisible by two positive numbers. Since there are infinitely many prime numbers, the statement is not sufficient.
Combining the two statements, we have p a prime number being divisible by 7, hence p must be 7. Therefore, both statements together are sufficient but neither statement alone is sufficient.
Ann deposited $3000 in her bank account at the beginning of the year. Determine the amount the funds accumulated to.
1. The bank offered 4.3% interest rate.
2. The amount was deposited for a period of 5 years.
Deposit (P) = 3000.
Accumulated amount = P + (1 + R/100)n where the variables have their usual meaning for compound interest and
Accumulated amount = P + (P × r/100 × n) where the variables have their usual meaning for simple interest.
In statement 1, r = 4.3 and P = 3000 but we are not given the value of n, hence we cannot find the accumulated amount. Further more, the statement does not give more information about the kind of interest offered, hence, it is not sufficient.
In statement 2, n = 5 and P = 3000 but we are not given the value of r, hence we cannot find the accumulated amount. Furthermore, the statement does not give more information about the kind of interest offered, hence, it is not sufficient.
Combining the two statements, P = 3000, r = 4.5 and n = 5 but, the details given are not sufficient since no specific type on interest is disclosed, therefore, we cannot apply the compound or simple interest formula with accuracy. Thus Statements (1) and (2) TOGETHER are NOT sufficient.
Find the mean of the data.
1. The data has 8 data values
2. The data is 3, 4, 5, 6, 4, 1, 0, 5
Mean = 1/n (∑x) where the symbols have their usual meaning.
In statement 1, n = 8, but ∑x is unknown, hence it is not sufficient.
In statement 2, ∑x = (3 + 4 + 5 + 6 + 4 + 1 + 0 + 5) = 28, and n =8 hence, mean = 28/8 = 3.5. The statement is sufficient.
Determine the volume of a cuboids.
1. The length is twice the width and the height is 4 inches.
2. The length is 6 inches.
Volume = length × width × height
In statement 1, let width be x, length = 2x and height = 4
Volume = x × 2x × 4 = 8x² cubic inches. Since it is in terms of unknown value, x, it is insufficient.
In statement 2, length = 6 inches but the width and height is unknown hence it is not sufficient to determine the volume.
Combining the two statements, length = 2x = 6 hence x = 3 inches.
width = 3 inches and height = 4 inches.
Volume = 3 × 4 × 6 = 72 cubic inches. Thus , BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Find the value of r if 4r + 2t = 14.
1. t = 2.
2. r > t.
4r + 2t = 14
In statement 1, t = 2, substituting for t in 4r + 2t = 14, we have
4r + 4 = 14; 4r = 10; r = 2 1/2. The statement is sufficient.
In statement 2, r > t , thus none of this is assigned a numeric value. Thisa implies that the solution of r in 4r + 2t = 14 will be a set of values. This implies that the statement is not sufficient.
Find the common difference of the arithmetic sequence.
1. The third term of the sequence is 1.428.
2. The first and the fifth terms of the sequence is 1 and 1.856.
a_{n} = a + (n  1)d
In statement 1, third term, a_{3} = a + (3  1)d = 1.428; a + 2d =1.428.
This is one equation with two unknowns hence we cannot determine the value of d. The statement is not sufficient.
In statement 2, a = 1 and a_{5} = a + (5  1)d = 1.856
Substituting for a, we have 1 + 4d = 1.856; d = 0.214.
The common difference is 0.214, therefore, the statement is sufficient.
Stephenson, a businessman bought an Iron box for $80. Determine his profit.
1. He made a 30% profit.
2. His selling price was $104.
Cost price = $80 is equivalent to 100%.
In statement 1, profit = 30%
= 30/100 × 80 = $24.
Profit = 104  80 = $24. The statement is sufficient.
In statement 2, selling price = 104.
selling price = cost price + profit; hence, profit =104  80 = $24. The statement is sufficient.
The ratio of water to alcohol in a 14 cup container is 2:5. Determine the new volume of the liquid in the container.
1. Water is increased by 14%.
2. Mixture whose ratio of water to alcohol is 4:5 is added to that in the container.
Ratio water : alcohol = 2 : 5.
To imply that water = 2/7 × 14 = 4 cups
Alcohol = 14  4 = 10 cups.
If water is increased by 14%, we have
new capacity of water =114/100 × 4 = 4.56 cups.
The new volume of the liquid is 10 + 4.56 cups. Hence the statement is sufficient.
Determine the area of a triangle A.
1. Triangle A and B are similar with a linear scale factor of 7 : 10.
2. B is larger than A.
In statement 1, linear scale factor of A : B = 7 : 10 hence the area scale factor = 49:100. However, we are not given the area of B hence we cannot determine the area of A. The statement is insufficient.
In statement 2, B is larger than A the ratio B/A is more than 1. Since no more information is given about the ratio or the area of either A or B, the statement is not sufficient.
Combining the two statements, Area scale factor = 49 : 100 hence, 49 corresponds to the smaller triangle and 100 to the larger triangle according to statement 2. However, we are not given the area of B hence we cannot determine the area of A. The statements together are not insufficient.
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