SAT MATH PROBLEMS ON EXPRESSIONS

Problem 1 :

Write an algebraic expression which represents the following verbal phrase.

"The denominator of a fraction exceeds the numerator by 5."

Solution :

Let x be the numerator.

The algebraic expression which represents the given verbal phrase :

x/(x + 5)

Problem 2 :

(1/x) ÷ (x + 3)

Which of the following is equivalent to the expression shown above? 

A) 1/[x(x + 3)]

B) x/(x + 3)

C) (x + 3)/x

D) x(x + 3)

Solution :

= (1/x) ÷ (x + 3)

= (1/x) ⋅ 1/(x + 3)

= 1/[x(x + 3)]

The correct answer choice is (A).

Problem 3 :

(1/m)² - 2(1/m)(1/n) + (1/n)²

Which of the following is equivalent to the expression shown above? 

A) (1/√m - 1/√n)⁴

B) (1/m - 1/n)²

C) 1/(m - n)²

D) 1/(m² - mn + n²)

Solution :

(1/m)² - 2(1/m)(1/n) + (1/n)²

The above expression is in the form of a² - 2ab + b².

Using the algebraic identity (a - b)² = a² - 2ab + b²,

(1/m)² - 2(1/m)(1/n) + (1/n)² = (1/m - 1/n)²

The correct answer choice is (B) (1/m - 1/n)².

Problem 4 :

(3x + 2y)²

The expression above can be written as ax² + bxy + cy²,  where a, b and c are constants.

What is the value of (a + b + c)?

Solution :

(3x + 2y)² :

= (3x + 2y)(3x + 2y)

= (3x)² + (3x)(2y) + (2y)(3x) + (2y)²

= 3²x² + 6xy + 6xy + 2²y²

= 9x² + 12xy + 4y²

Comparing

ax² + bxy + cy²

and

 9x² + 12xy + 4y²

we get a = 9, b = 12 and c = 4.

a + b + c = 9 + 12 + 4

= 25

Problem 5 :

On Sunday, Jack read x pages every 15 minutes for 5 hours, and Lily read y pages every 30 minutes for 4 hours. Which of the following represents the total number of pages read by Jack and Lily on Friday?

Solution :

Given : Jack read x pages every 15 minutes for 5 hours.

15 minutes ----> x pages

1 hour ----> 4x pages

5 hours ----> 5(4x) = 20x pages

Given : Lily read y pages every 30 minutes for 4 hours.

30 minutes ----> y pages

1 hour ----> 2y pages

4 hours ----> 4(2y) = 8y pages

Total number of pages read by Jack and Lily on Sunday :

= 20x + 8y

Problem 6 :

If a = x² - 5x + 2 and b = 3x³ + 4x² - 6, what is 3a - b in terms of x?

Solution :

3a - b = 3(x² - 5x + 2) - (3x³ + 4x² - 6)

= 3x² - 15x + 6 - 3x³ - 4x² + 6

Group the like terms together.

= -3x³ + (3x² - 4x²) - 15x + (6 + 6)

Combine the like terms.

= -3x³ + (-x²) - 15x + 12

= -3x³ - x² - 15x + 12

Problem 7 :

If p + q = -5 and p - q = -12, find the value of p² - q².

Solution :

p + q = -5 ----(1)

p - q = -12 ----(2)

Multiply (1) and (2).

(p + q)(p - q) = (-5)(-12)

p² - pq + pq - q² = 60

p² - q² = 60

Problem 8 :

If a² + b² = 13 and ab = 6, find the value of (a + b) such that (a + b) > 0. 

Solution :

(a + b)² = (a + b)(a + b)

(a + b)² = a² + ab + ab + b²

(a + b)² = a² + 2ab + b²

(a + b)² = a² + b² + 2ab

Substitute a² + b² = 13 and ab = 6.

(a + b)² = 13 + 2(6)

(a + b)² = 13 + 12

(a + b)² = 25

Taking square root on both sides.

√(a + b)² = √25

a + b = ±5

a + b = 5  or  a + b = -5

Since (a + b) > 0,

a + b = 5

Problem 9 :

A grocery store uses crates to store a total of 36a apples and 24w watermelons. Each crate can be used to store either 12 apples or 6 watermelons. Write the expression which gives the total number of crates the grocery store uses to store apples and watermelons.

Solution :

Given : Each crate can be used to store either 12 apples or 6 watermelons.

There are 36a apples and 24w watermelons.

Number of crates required to store 36a apples :

= 36a/12

= 3a

Number of crates required to store 24w apples :

= 24w/6

= 4w

Total number of crates required to store 36a apples and 24w watermelons :

= 3a + 4w

Problem 10 :

(m + n + 1)(m + n - 1)

Which of the following is equivalent to the expression shown above?

A) m² + 2mn + n² - 1

B) m² - 2mn + n² - 1

C) m² - n² - 1

D) m² + 2m + n² + 2n - 1

Solution :

= (m + n + 1)(m + n - 1)

Let a = m + n.

= (a + 1)(a - 1)

Using the algebraic identity a² - b² = (a + b)(a - b),

= a² - 1²

= a² - 1

Replace 'a' by (m + n).

= (m + n)² - 1

Using the algebraic identity (a + b) = a² + 2ab + b²,

= m² + 2mn + n² - 1

The correct answer choice is (A).

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