← Back to the Expression Structure worksheet
Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Standards: A-SSE.A.1a, A-SSE.A.1b, N-RN.B.3 · Level: Grade 9 Algebra 1
Problem 2 Coefficients and constant
The coefficient of a is 8; of b it is −5 (the sign belongs to the term). The constant term is 2.
8, −5 and constant 2
Problem 3 Factors of a term
The term is a product of 7, x and y.
7, x, y
Problem 4 Classify points on a line
1/2 is a quotient of integers, so it is rational. √2 and π have decimals that never end or repeat.
B (√2) and C (π) are irrational.
Problem 5 Factors of a product
The expression is a product of three factors: 2, (x + 5) and (x − 1).
3 factors
Problem 6 Gym membership
The term 35m is 35 dollars for each month. The constant 20 does not change with m, so it is a one-time sign-up fee of 20 dollars.
35 = cost per month; 20 = fixed sign-up fee
Problem 7 Interpret an exponential
At t = 0 the amount is 1200(1.05)0 = 1200, the starting deposit. Each year multiplies by 1.05, so the balance grows 5% per year.
1200 = starting deposit; 1.05 = yearly growth factor (5% growth)
Problem 8 Factored view
The factor (n + 3) is the number of people: n students plus 3 chaperones. The factor 4 is tickets per person.
(n + 3) = people; 4 = tickets per person
Problem 9 Rational plus irrational
Suppose 5 + √3 were rational. Subtracting the rational number 5 would give a rational result, but √3 is irrational. That is a contradiction.
5 + √3 is irrational.
Problem 10 Rational times irrational
If (2/3)√5 were rational, multiplying by the rational number 3/2 would make √5 rational, which it is not.
Irrational
Problem 11 Two irrationals, rational sum
The number 3 − √7 is irrational (rational minus irrational). The sum is √7 + (3 − √7) = 3.
The sum is 3, a rational number.
Problem 12 Product of two roots
√2 · √8 = √16 = 4.
Check: √8 ≈ 2.828 and 1.414 × 2.828 ≈ 4.
Rational (equal to 4)
Problem 13 Draining tank
80 is the starting amount; −6t is the total drained after t minutes (6 gallons per minute).
When t = 5: 80 − 6(5) = 80 − 30 = 50.
50 gallons
Problem 14 Evaluate a growth expression
(1.03)2 = 1.0609. Then 800 × 1.0609 = 848.72.
After 2 years, 800 dollars grows to $848.72.
Problem 15 Savings account
Growth factor: 1 + 0.04 = 1.04. Then 1.04³ = 1.124864 and 2,500 × 1.124864 = 2,812.16.
A = $2,812.16, with growth factor 1.04
Problem 16 Bike rental plans
(a) Shop A: fixed 15 versus 3.
(b) Shop B: 8 per hour versus 6.
(c) A: 15 + 6(4) = 39. B: 3 + 8(4) = 35.
(a) Shop A (b) Shop B (c) $39 and $35
Problem 17 Garden perimeter
2(w + w + 8) = 2(2w + 8) = 4w + 16.
The term 4w counts the four widths’ worth of fence; 16 is the two extra 8-foot stretches. For w = 6: 4(6) + 16 = 40.
Perimeter = 4w + 16; 40 feet when w = 6
Problem 18 Circular pond
Circumference: 2π(2.5) = 5π. Area: π(6.25) = 6.25π.
Each is a nonzero rational number times π, so each is irrational.
5π ≈ 15.71 and 6.25π ≈ 19.63.
Both are irrational: about 15.71 m and 19.63 m²
Problem 19 Car value
(a) The factor 0.85 keeps 85% of the value, so 15% is lost each year.
(b) 0.85³ = 0.614125. Then 24,000 × 0.614125 = 14,739.
(a) 15% per year (b) $14,739
Problem 20 Compare two accounts
1.0158 ≈ 1.126493, so A ≈ $1,126.49.
1.034 = 1.12550881, so B ≈ $1,125.51.
Difference: 1,126.49 − 1,125.51 = 0.98.
Account A is larger by about $0.98