If Amy took 30% and Beth took 20%, then Charlie took 50% of the pizza.
Let P denote the size of the total pizza.
Amount Amy ate = (3/4) (0.3 P) = 0.225 P.
Amount Charlie ate = (2/5) (0.5 P) = 0.2 P.

Be careful! Note that (61 - 79) is a negative number.
Quantity A is negative because it is the product of a positive number and a negative number.
Quantity B is positive because it is the product of two negative numbers.

If a, b, c and d are four consecutive integers,
then b = a + 1 ; c = a + 2 ; and d = a + 3.
Quantity A = (a + d)/2 = (a + a + 3)/2 = (2a + 3)/2.
Quantity B = (b + c)/2 = (a + 1 + a + 2)/2 = (2a + 3)/2.

You could substitute values for z between 2 and 4 to compare.
An alternative simple strategy is to divide both Quantity A and B by π²z².
Then Quantity A = z and Quantity B = π.
We note that Quantity A lies between 2 and 4, whereas Quantity B equals about 3.14

(y − z)² = (y − z)(y − z) = y² + z² − 2yz.
Since yz is negative, −2yz is positive. So, Quantity A > Quantity B.

Dividing both sides by 11/3 gives y⁴/y² = 1 or y⁴ = y².
Now, dividing both sides by y² gives y² = 1.
So, y can equal 1 or -1.

When y = 1, Quantity A equals Quantity B.
When y > 1, Quantity A is greater. When y < 1, Quantity B is greater.

Area of rectangle = Length x Breadth
Perimeter of rectangle = 2 (Length + Breadth)
If the area of the rectangle is 33, then there are many possible values for the length and breadth as given below.
Length = 11, Width = 3 and Perimeter = 28.
Length = 33, Width = 1 and Perimeter = 68.
Length = 6, Width = 5.5 and Perimeter = 23.

Expanding the left-hand side will give a cubic polynomial and the resulting cubic equation will be difficult to solve. The following strategy may be
adopted.
If n + 2 = 6, then n = 4, n + 4 = 8 and 4 x 6 x 8 = 192.
Since the product must be 480, n + 2 must be larger than 6.

Let Bill stitch b shirts per day. Then, Andrew stitches 3b shirts per day.
Let Andrew charge a dollars per shirt. Then, Bill charges 1.4a dollars per shirt.
Amount Andrew earns in 8 days = 8 (3b) a = 24 ab.
Amount Bill earns in 15 days = 15 b (1.4a) = 21 ab.

12p = 12 (3z/4) = 9z.
q = 4r/5 = 4z/6.
12q = 12 (4z/6) = 8z.
If z = 0, then 9z = 8z = 0.
If z > 0, then 9z > 8z.
If z < 0, then 9z < 8z.

20! / 17! = (20 x 19 x 18 x 17 x 16 x ...)/(17 x 16 x 15 x ...) = 20 x 19 x 18.
80! / 78! = (80 x 79 x 78 x 77 x ...)/(78 x 77 x 76 x ...) = 80 x 79 = 20 x 4 x 79.
Now, 19 x 18 > 4 x 79.

When a > 1, then a³ > a².
When a = 1, then a³ = a² = 1.
When 0 < a < 1, then a³ < a².
For example, (1/2)³ = 1/8, (1/2)² = 1/4, and 1/8 < 1/4.

On dividing by m² (a positive quantity),
Quantity A = m and Quantity B = -1.
When m = -1, then Quantity A equals Quantity B.
When -1 < m < 0, then Quantity A is greater.
When m < -1, then Quantity B is greater.
For example, (-1/2)³ = -1/8, -(-1/2)² = -1/4, and -1/8 > -1/4.

Every third integer is a multiple of 3 and every fifth integer is a multiple of 5. The number of multiples of 3 will typically be greater than the number
of multiples of 5 in a large enough interval (if y is small). However, this will not be the case if y is large, e.g., if y = 199, then there is no multiple of 3 between 199 and 200, and there is only one multiple of 5.

BC is a two-digit number, so it is less than 100.
BC + BC gives a three-digit number which is therefore between 100 and 198.
So, C in CAA must represent 1.
Adding the digits in the units place gives 1 + 1 = 2. So, A must represent 2.
Finally, C + C = 12 and therefore C must represent 6.

The average of the three angles of a triangle is the sum divided by 3.
The sum of the angles of any triangle is always 180^o.
So, the average is 60^o irrespective of the kind of triangle.

Let h be the height of the equilateral triangle.
By Pythagoras' Theorem, h² = a² - (a/2)² = 3a²/4.
So, the height h is less than a.
Area of triangle = (base x height)/2.
2 (Area of triangle) = base x height < a² = Area of square.

If r is the radius of the small circle, then 1.4r is the radius of the large circle.
Area of small circle = π r².
Area between circles = π (1.4r)² − π r² = (1.4² − 1) π r² = 0.96 π r².

3 + 13 = 16 and 5 + 11 = 16 are the two possibilities.
So, n can be either 11 or 13.