ℚ
Recognize rational numbers
Connect integers, fractions, mixed numbers, terminating decimals, and repeating decimals.
Worked examples + 20 practice questions
Grade 7 Rational Numbers Review
Review positive and negative fractions and decimals, number-line reasoning, all four operations, absolute value, decimal forms, and mixed real-world problems.
This guide follows the Grade 7 Number System strand for operations with rational numbers and real-world problem solving. View the official mathematics standards. State and district sequences may differ.
The Number System
Six connected rational-number skills
Grade 7 extends whole-number and fraction understanding to positive and negative rational numbers.
↔
Use the number line
Interpret direction, opposites, absolute value, addition, subtraction, and distance.
+
Add and subtract
Combine signed fractions and decimals and rewrite subtraction as addition of the opposite.
×
Multiply and divide
Determine the sign, calculate accurately, and interpret products and quotients.
0.3̅
Convert decimal forms
Use long division to identify terminating and repeating rational decimals.
Σ
Solve mixed problems
Apply all four operations, order of operations, estimation, units, and context.
Review map
What this guide covers
Concept 1
A rational number can be written as a fraction
A number is rational when it can be written as a/b, where a and b are integers and b is not zero.
Rational examples
- Integers: −8 = −8/1 and 5 = 5/1
- Fractions: −3/4, 7/10, and 11/6
- Mixed numbers: −2 1/3 and 4 5/8
- Terminating decimals: 0.875 and −1.2
- Repeating decimals: 0.333… and 0.4333…
Equivalent forms
| Fraction | Decimal | Type |
|---|---|---|
| 3/4 | 0.75 | Terminating |
| 7/8 | 0.875 | Terminating |
| 1/3 | 0.333… | Repeating |
| 13/30 | 0.4333… | Repeating |
Denominator restriction: The denominator cannot be zero because division by zero is undefined.
Concept 2
Use direction and distance on the number line
Positive direction is to the right, negative direction is to the left, and absolute value measures distance from zero.
A rational-number number line
-5-4-3-2-1012345p = −4q = 2.5
The distance between −4 and 2.5 is |2.5 − (−4)| = |6.5| = 6.5 units.
Opposites
A number and its opposite are the same distance from zero but lie in opposite directions.
Absolute value
Absolute value is distance, so it is never negative.
Subtraction distance
The distance between p and q is |p − q| or |q − p|.
Concept 3
Add by combining movement; subtract by adding the opposite
The identity p − q = p + (−q) gives one consistent method for subtraction with fractions, decimals, and integers.
Addition
- Same signs: add magnitudes and keep the common sign.
- Different signs: subtract the smaller magnitude from the larger magnitude.
- Use the sign of the number with the greater magnitude.
- Use a common denominator before adding fractions.
Subtraction
- Rewrite subtraction as addition of the opposite.
- Keep the first number unchanged.
- Change the subtraction sign to addition.
- Replace the second number by its opposite.
- Then apply the addition rules.
p − q=p + (−q)
Concept 4
Determine the sign before multiplying or dividing
The same sign rule applies to multiplication and division, but the numerical procedure still depends on whether the values are fractions, decimals, or mixed numbers.
Fractions
- Convert mixed numbers to improper fractions.
- Multiply numerators and denominators.
- For division, multiply by the reciprocal.
- Simplify the final fraction.
Decimals
- First decide the sign.
- Calculate using positive magnitudes.
- Place the decimal accurately.
- Estimate to check the size.
More than two factors
- An even number of negative factors gives a positive product.
- An odd number of negative factors gives a negative product.
- Group convenient factors when using properties.
Concept 5
Connect fractions, decimals, and order of operations
A rational number written as a decimal terminates or eventually repeats. Mixed expressions still follow the usual order of operations.
Terminating decimal
The long-division process ends.
7/8 = 0.875
Repeating decimal
A digit or block of digits repeats indefinitely.
13/30 = 0.4333…
Mixed expression
Parentheses, multiplication or division, then addition or subtraction.
−4.5 + 2(1.75 − 3.25)
Three checks before accepting an answer
- Sign check: should the result be positive or negative?
- Magnitude check: is the answer a reasonable size?
- Form check: should the answer be a fraction, decimal, mixed number, distance, or contextual quantity?
Step by step
Eight worked rational-number examples
State the sign or operation meaning first, then calculate and check the result.
1. Add numbers with different signs
Compute −8 + 13.
- The numbers have different signs.
- Subtract magnitudes: 13 − 8 = 5.
- The positive number has the greater magnitude.
Answer: 5
2. Rewrite subtraction
Compute −3.5 − 2.25.
- Rewrite as −3.5 + (−2.25).
- The signs are the same, so add the magnitudes.
- 3.5 + 2.25 = 5.75 and keep the negative sign.
Answer: −5.75
3. Add signed fractions
Compute −3/5 + 7/10.
- Rewrite −3/5 as −6/10.
- Add: −6/10 + 7/10 = 1/10.
- The positive fraction has the greater magnitude.
Answer: 1/10
4. Multiply signed fractions
Compute (−3/4)(8/9).
- Different signs give a negative product.
- Multiply: (3 × 8)/(4 × 9) = 24/36.
- Simplify 24/36 to 2/3.
Answer: −2/3
5. Divide signed fractions
Compute (−5/6) ÷ (10/9).
- Different signs give a negative quotient.
- Multiply by the reciprocal: (−5/6)(9/10).
- Simplify −45/60 to −3/4.
Answer: −3/4
6. Convert a fraction to a decimal
Write 13/30 as a decimal.
- Divide 13 by 30.
- The quotient begins 0.4 with remainder 1.
- The digit 3 then repeats.
Answer: 0.4333…
7. Apply order of operations
Evaluate −4.5 + 2(1.75 − 3.25).
- Parentheses: 1.75 − 3.25 = −1.5.
- Multiply: 2(−1.5) = −3.
- Add: −4.5 + (−3) = −7.5.
Answer: −7.5
8. Interpret a real-world change
The temperature is 6.5°C and drops 9.75°C.
- A drop is subtraction: 6.5 − 9.75.
- Rewrite as 6.5 + (−9.75).
- The result is 3.25 units below zero.
Answer: −3.25°C
Error prevention
Eight common rational-number mistakes
Adding signs instead of values
The signs tell direction; they are not separate numbers to combine.
Forgetting to add the opposite
Subtraction should be rewritten before applying addition rules.
Using unequal denominators
Fractions require a common denominator for addition or subtraction.
Applying sign rules to addition
The same-sign and different-sign multiplication rule does not replace addition reasoning.
Forgetting the reciprocal
Fraction division means multiply by the reciprocal of the divisor.
Losing a negative sign
Decide and write the sign before multiplying or dividing magnitudes.
Ignoring order of operations
Parentheses and multiplication or division must be completed before addition or subtraction.
Reporting negative distance
Distance and absolute value are never negative.
Independent practice
20 Grade 7 rational numbers questions
Show the sign decision, rewrite subtraction where helpful, and simplify exact fractions.
A. Addition, subtraction, and distance
- Compute: −7 + 12.
- Compute: 4.5 − 9.25.
- Compute: −3/5 + 7/10.
- Compute: −2 1/4 − 1 2/3.
- Find the distance between −4.75 and 2.5 on a number line.
B. Multiplication and division
- Compute: (−3/4)(8/9).
- Compute: −2.4 × 3.5.
- Compute: (−5/6) ÷ (10/9).
- Compute: (−1.2)(3.5)(−2).
- Compute: 7/9 ÷ (−14/15).
C. Decimal forms and mixed expressions
- Write 7/8 as a decimal and state whether it terminates or repeats.
- Write 13/30 as a decimal and state whether it terminates or repeats.
- Write −11/16 as a decimal.
- Evaluate: −4.5 + 2(1.75 − 3.25).
- Evaluate: 6 − [2.5 − (−1.75)].
D. Real-world rational-number problems
- The temperature is 6.5°C and drops 9.75°C. Find the new temperature.
- A bank account contains $45.00. A payment of $72.50 is withdrawn. Find the new balance.
- A diver is at −18.5 m, rises 7.25 m, then descends 12.75 m. Find the final position.
- An elevator starts at −125 ft, rises 68 ft, then descends 42 ft. Find the final elevation.
- A player's score changes by −3.75 points, +5.5 points, and −2.25 points. Find the total change.
Complete all four sections before opening the answer key.
Go to the Answer KeyCheck your work
Rational numbers answer key
Use the explanation to identify whether the error involved sign, operation, fraction procedure, decimal placement, or context.
Question 15
The signs differ, so subtract 12 − 7 and use the positive sign.
Question 2−4.75
4.5 − 9.25 = 4.5 + (−9.25).
Question 31/10
−3/5 = −6/10, so −6/10 + 7/10 = 1/10.
Question 4−47/12, or −3 11/12
−2 1/4 = −27/12 and −1 2/3 = −20/12; add to get −47/12.
Question 57.25 units
|2.5 − (−4.75)| = |7.25|.
Question 6−2/3
Different signs give a negative product; 24/36 simplifies to 2/3.
Question 7−8.4
Different signs give a negative product; 2.4 × 3.5 = 8.4.
Question 8−3/4
Multiply −5/6 by the reciprocal 9/10 and simplify.
Question 98.4
Two negative factors produce a positive product.
Question 10−5/6
7/9 × (−15/14) simplifies to −5/6.
Question 110.875; terminating
Long division ends after three decimal places.
Question 120.4333…; repeating
The digit 3 repeats after the initial 4.
Question 13−0.6875
11 ÷ 16 = 0.6875, then apply the negative sign.
Question 14−7.5
Parentheses give −1.5; multiply by 2, then add to −4.5.
Question 151.75
2.5 − (−1.75) = 4.25; then 6 − 4.25.
Question 16−3.25°C
6.5 − 9.75 = −3.25.
Question 17−$27.50
45.00 − 72.50 = −27.50.
Question 18−24 m
−18.5 + 7.25 − 12.75 = −24.
Question 19−99 ft
−125 + 68 − 42 = −99.
Question 20−0.5 points
−3.75 + 5.5 − 2.25 = −0.5.
Choose the next review from the error pattern
- Questions 1–5: review number-line direction, adding the opposite, common denominators, and absolute-value distance.
- Questions 6–10: review sign reasoning, fraction multiplication, reciprocals, and decimal multiplication.
- Questions 11–15: review long division, terminating and repeating decimals, and order of operations.
- Questions 16–20: review translating gains, losses, rises, drops, deposits, withdrawals, and elevation changes.
Focused review
A five-session rational numbers plan
Number line and addition
Opposites, absolute value, distance, and signed addition.
Subtraction
Rewrite as adding the opposite with integers, fractions, and decimals.
Multiplication and division
Sign reasoning, reciprocals, simplification, and estimation.
Decimal forms and order
Long division, terminating or repeating decimals, and mixed expressions.
Real-world transfer
Complete the 20-question set, classify errors, and retry weak types.
Frequently asked questions
Grade 7 rational numbers FAQ
What is a rational number in Grade 7 math?
A rational number is any number that can be written as a fraction a over b, where a and b are integers and b is not zero. Integers, terminating decimals, and repeating decimals are rational.
What rational-number operations are taught in Grade 7?
Grade 7 students add, subtract, multiply, and divide positive and negative rational numbers, explain the meaning of the operations, and solve mathematical and real-world problems.
Why is subtracting a number the same as adding its opposite?
Subtraction can be rewritten as p minus q equals p plus negative q. This connects subtraction to addition and allows one consistent number-line model.
How can a student remember multiplication and division signs?
A product or quotient is positive when the two signs are the same and negative when the signs are different. Students should also connect the rule to patterns and the distributive property rather than memorizing it alone.
How do you know whether a rational decimal terminates or repeats?
Every rational number has a decimal expansion that terminates or eventually repeats. Long division can reveal the pattern.
How should a student study Grade 7 rational numbers?
Practice one operation at a time, explain the sign before calculating, use fractions and decimals in the same session, check results with estimation, and finish with mixed real-world problems.
Standards reference and scope
This guide is organized around the Grade 7 Number System standards for adding, subtracting, multiplying, and dividing rational numbers; interpreting operations; converting rational numbers to decimal form; and solving mathematical and real-world problems. Read the official Common Core mathematics standards. GradeCove is an independent educational publisher and is not endorsed by the Common Core State Standards Initiative.
Need complete Grade 7 assessment and practice?
Use the full revision pack for a diagnostic test, three 40-question practice papers, worked solutions, a study plan, and domain tracking.