Here's how our practice questions align with the logic and structure of the Official GMAT Focus Exam:

• GMAT-Style Precision: Our practice questions are meticulously designed to mirror the logic and structure of Official GMAT Focus questions, ensuring that students engage with problems that closely resemble those on the actual exam.
• Complex Problem-Solving: We emphasize the same level of complexity found in OG questions, allowing students to develop the critical reasoning and quantitative problem-solving skills.
• Balanced Difficulty: Our question sets are balanced to reflect both medium and hard difficulty levels, similar to the adaptive nature of the GMAT Focus, ensuring a comprehensive preparation experience.

You started by correctly identifying that the original price ratio of the hollow pine door to the solid oak door is 1:2, with prices of $40 and $80, respectively.

However, the mistake happens when you try to adjust the ratio after applying the 25% discount. You calculated that the final ratio should be 1:1.75 and that this implies a new price of $70 for the solid oak door. This step is where things go wrong.

Here’s why:

1. Discount Application: The problem states that Cheryl received a 25% discount on the solid oak doors. This means you should reduce the original $80 price by 25%, which gives you $60, not $70. The correct calculation is:
80 × (1 - 0.25) = 80 × 0.75 = 60 dollars
So, the price after the discount is fixed at $60.

2. Incorrect Ratio Adjustment: The ratio of 1:1.75 that you used isn't necessary here. The discount simply reduces the price to $60, and we should work with that exact price. Adjusting to $70 based on the ratio leads to an incorrect total cost because it doesn’t reflect the actual 25% discount.

Correct Total Price:
When you use the correct discounted price of $60 for the solid oak doors, the total cost becomes:
5 × 40 + 6 × 60 = 200 + 360 = 560 dollars

I hope this helps :)

Understand the problem
We need to find integer solutions for the equation (a−b)^2 + 8b^2 = 108.

Substitute different values of b and solve for a
Let k = a - b. Then the equation becomes k^2 + 8b^2 = 108.

Test values of b
b = 0: k^2 + 0 = 108 -> No solution (k^2 = 108 is not a perfect square)
b = 1: k^2 + 8 = 108 -> k^2 = 100 -> k = ±10 -> (a, b) = (11, 1), (-9, 1)
b = 2: k^2 + 32 = 108 -> k^2 = 76 -> No solution (k^2 = 76 is not a perfect square)
b = 3: k^2 + 72 = 108 -> k^2 = 36 -> k = ±6 -> (a, b) = (9, 3), (-3, 3)
b = 4: k^2 + 128 = 108 -> No solution (k^2 = -20 is not possible)
b = -1: k^2 + 8 = 108 -> k^2 = 100 -> k = ±10 -> (a, b) = (9, -1), (-11, -1)
b = -3: k^2 + 72 = 108 -> k^2 = 36 -> k = ±6 -> (a, b) = (3, -3), (-9, -3)

Count the solutions
The ordered pairs (a, b) are:
(11, 1), (-9, 1), (9, 3), (-3, 3), (9, -1), (-11, -1), (3, -3), (-9, -3)

Conclusion
There are 8 ordered pairs (a, b).

First, we can note that 500% of a number is equivalent to 5 times that number, and 400% of a number is equivalent to 4 times that number. Next, with these facts in mind, we can create two equations:

p = 5q (equation 1)

q = 4z (equation 2)

We can substitute 4z for q in equation 1:

p = 5(4z)

p = 20z (equation 3)

Now, to find out what percent z is of p, we can divide these two quantities and multiply the result by 100:

z/p ✕ 100

Finally, we can simplify this expression if we use equation 3, plugging in 20z for p:

z/p ✕ 100 = z/20z ✕ 100 = 1/20 ✕ 100 = 100/20 = 5

Thus, z is 5% of p.

Understand the problem
Given: a, b, c in G.P.
Median is a, and b = 10
|a + b + c| = 15
a > c

Express G.P. terms
Since b is the middle term: a/r, 10, ar
|(a/r) + 10 + ar| = 15

Solve the absolute value equation
Case 1: (a/r) + 10 + ar = 15
(a/r) + ar = 5
a(r + 1/r) = 5

Case 2: (a/r) + 10 + ar = -15
(a/r) + ar = -25
a(r + 1/r) = -25

Solve for r
Let's solve case 1 since |a + b + c| should be positive.
ar = k, where k is the common ratio.

a(1/r + r) = 5
10(1/r + r) = 5
1/r + r = 0.5
Let r^2 + 2 = k, then
2 + 1/r = 5

Determine the product of first 4 terms
First 4 terms: a/r, 10, ar, a(r^2)

Since a > c:
a > ar
ar = 10, hence, c = a

Product of first 4 terms: (a/r) * 10 * ar * a

Factorize the denominator
2x^2 + 8x + 8
= 2(x^2 + 4x + 4)
= 2(x + 2)^2

Simplify the fraction
(2x + 4) / (2(x + 2)^2)
= 2(x + 2) / 2(x + 2)(x + 2)
= 1 / (x + 2)

Conclusion
The equivalent expression is 1 / (x + 2)

Understand the problem
We need to form four-digit numbers using the digits {1, 2, 3, 4, 5, 5, 6, 6}.
Digits can be repeated as given in the set, but the numbers must be distinct.

Calculate the number of permutations
Total digits: 8
We need to choose 4 out of these 8 digits.

Consider repetitions
For each selection of 4 digits, we need to consider different cases for repetitions:
1. All digits are unique
2. One digit is repeated

Case 1: All digits are unique
Select 4 digits from {1, 2, 3, 4, 5, 6}
Number of ways to select 4 out of 6 = C(6, 4) = 15
Number of permutations of 4 unique digits = 4! = 24
Total = 15 * 24 = 360

Case 2: One digit is repeated
Choose 3 out of {1, 2, 3, 4, 5, 6}
Number of ways to select 3 out of 6 = C(6, 3) = 20
Number of permutations with one repetition = 4!/2! = 12
Total = 20 * 12 = 240

Sum the cases
Total distinct four-digit numbers = 360 + 240 = 600

Understand the problem
We have two equations: 2x = 10 and 3y = 10.
We need to find the value of 12xy.

Solve for x and y
2x = 10
x = 10 / 2
x = 5

3y = 10
y = 10 / 3
y ≈ 3.33

Calculate 12xy
12xy = 12 * 5 * (10 / 3)
12xy = 12 * 5 * 3.33
12xy = 200

Understand the problem
We need to minimize x^2 + 2xy + y^2 given 2x + 9y = 57.

Express y in terms of x
2x + 9y = 57
y = (57 - 2x) / 9

Substitute y in the expression to be minimized
f(x, y) = x^2 + 2xy + y^2

Find integer solutions for y
x must be chosen such that (57 - 2x) / 9 is an integer.
Test values for x:
x = 3: y = (57 - 6) / 9 = 51 / 9 = 5.67 (not an integer)
x = 12: y = (57 - 24) / 9 = 33 / 9 = 3.67 (not an integer)
x = 21: y = (57 - 42) / 9 = 15 / 9 = 1.67 (not an integer)
x = 30: y = (57 - 60) / 9 = -3 / 9 = -1/3 (not an integer)
x = 39: y = (57 - 78) / 9 = -21 / 9 = -2.33 (not an integer)

Verify integer values for y
Find values of x such that y is an integer:
x = -3, y = 7

Compute x^2 + 2xy + y^2
For x = -3, y = 7:
f(-3, 7) = (-3)^2 + 2(-3)(7) + (7)^2
= 9 - 42 + 49
= 16

Conclusion
The least possible value of x^2 + 2xy + y^2 is 16.

Understand the problem
The tank is initially filled to 3/4 of its capacity.
After adding 6 gallons, it is filled to 9/10 of its capacity.

Set up the equation
Let the capacity of the tank be C gallons.
Initial amount of gasoline = (3/4)C
After adding 6 gallons, the amount of gasoline = (3/4)C + 6

This is equal to 9/10 of the tank's capacity:
(3/4)C + 6 = (9/10)C

Solve for C
(3/4)C + 6 = (9/10)C
Multiply through by 20 to clear fractions:
20 * (3/4)C + 20 * 6 = 20 * (9/10)C
15C + 120 = 18C

Subtract 15C from both sides:
120 = 3C

Divide both sides by 3:
C = 40

Conclusion
The tank holds 40 gallons.