Fractions Practice Questions with Solutions (Hard)

Fractions are a way of representing parts of a whole. They express a number in terms of a numerator and a denominator, where:

• Numerator: The top number, represents how many parts we are considering.
• Denominator: The bottom number, represents the total number of equal parts that make up the whole.

Types of Fractions:

1. Proper Fraction: The numerator is smaller than the denominator (e.g., 3/4).
2. Improper Fraction: The numerator is equal to or larger than the denominator (e.g., 5/4).
3. Mixed Number: A whole number combined with a proper fraction (e.g., 1\frac{1}{2}).

Read More: Fun Facts about Fractions

Solved Questions on Fractions (Hard)

Question 1: Solve for x: 2/(x + 3) + 4/(x + 5) = 1/2

Solution:

Find a common denominator for the fractions on the left-hand side. The denominators are x + 3 and x + 5. The common denominator will be (x + 3)(x + 5).

Multiply and divide the first fraction with (x +5)
Multiply and divide the second fraction with (x + 3)

Combine the fractions on the left-hand side
2(x + 5) + 4(x + 3)/(x + 3) (x +5) ​= 1/2​
2x + 10 + 4x + 12/(x + 3)(x + 5) = 1/2​
6x + 22/(x + 3)(x +5) = 1/2

Cross-multiply the fractions:
2(6x + 22) = (x + 3)(x + 5)
12x + 44 = x² + 8x + 15
x² + 8x + 15 − 12x − 44 = 0
x2 − 4x − 29 = 0
Solve the quadratic equation x² − 4x − 29 = 0 using the quadratic formula:
x = 2 + √33 ​or x = 2 − √33​

Question 2: Solve for y: (y + 1)/3 − 2/(y − 4) = 5/6

Solution:

(y + 1)/3 − 2/(y − 4) = 5/6
(y + 1)/3 = 5/6 + 2/y - 4

Find the common denominator on the right-hand side. The common denominator between 6 and y − 4y is 6(y − 4), so:
5/6 + 2/y - 4 = 5(y - 4)/6(y - 4) + 12/6(y - 4)
5(y − 4) + 12/6(y - 4) = 5y - 20 + 12/6(y - 4)
= 5y -8/6(y - 4)
y + 1/3 = 5y -8/6(y -4)​

Upon Cross-multiply:
6(y − 4)(y + 1) = 3(5y − 8)
6(y² + y − 4y − 4) = 15y -24
6y² + 18y -24 =15y - 24
6y² − 33y = 0
3y(2y − 11) = 0
y = 0 ore 2y - 11 =0

y =0 or y =11/2

Question 3: In a class of 90 students, 1/3rd of the students do not like cricket. How many students like cricket?

Solution:

Fraction of students that do not like cricket = 1/3

Fraction of student that like cricket = 1 – 1/3
= (3 – 1)/3
= 2/3rd students like cricket.

Number of students that like cricket = 2/3 × 90
= (2 × 30)
= 60 

Therefore, 60 students like cricket.

Question 4: If a recipe needs 3/4 cup of sugar and you want to make twice the quantity mentioned in the recipe, how much sugar do you need?

Solution: 

Sugar needed for recipe = 3/4 cup

Sugar needed for half the recipe = 2 × 3/4
Required sugar = 2× 3/4 = 6/4

Therefore, we need 6/4 cup of sugar.

Question 5: Samantha ordered a large pizza for her party. She cut the pizza into 8 equal slices. She ate 3/8 of the pizza, and her friend Chris ate 2/8 of the pizza. How much of the pizza is left?
Solution:

Samantha ate 3/8 of the pizza, and Chris ate 2/8. So the total amount of pizza eaten is:
3/8 + 2/8 = 5/8

The total pizza is 1 (or 8/8), so to find how much is left, subtract the amount eaten from 1:
1 − 5/5 ​= 8/8 ​− 5/8 ​= 3/8​

So, 3/8 of the pizzais left.

Question 6: Alex has a bag of 120 candies. He gives 2/5 of the candies to his friend Ben. How many candies does Alex give to Ben?

Solution:

Find 2/5 of 120.
2/5 × 120 = 2 × 120/5 = 240/5 = 48
So, Alex gives 48 caneis to Ben.

Question 7: Factor the expression and simplify the fraction: x² + 5x + 6/x² + 7x + 12

Solution:

Factor the numerator: x² + 5x + 6:
The factors of 6 that add up to 5 are 2 and 3. So, we can factor the numerator as: x² + 5x + 6 = (x + 2)(x + 3)

Factor the denominator x² + 7x + 12x²:
The factors of 12 that add up to 7 are 3 and 4. So, we can factor the denominator as: x² + 7x + 12x² = (x + 3)(x + 4)

Rewrite the fraction with factored expressions:
(x + 2)(x + 3)/(x + 3)(x + 4)

The simplified expression is:
x + 2/x + 4

Question 8: Factor the numerator and denominator, then simplify: x² − 9/x² + 5x + 6
Solution:

Factor the numerator x² − 9
This is a difference of squares, which factors as: x² − 9 = x² -3² (x − 3)(x + 3)

Factor the denominator x² + 5x + 6.
The factors of 6 that add up to 5 are 2 and 3. So, we can factor the denominator as: x² + 5x + 6 = (x + 2)(x + 3)

Rewrite the fraction with factored expressions:
(x − 3)(x + 3)/(x + 2)(x + 3)

The simplified expression is:
x − 3/x + 2

Practice Questions on Frations (Hard)

Question 1: Solve for x: 3/(x + 2) + 5/(x + 4) = 1/3

Question 2: Solve for y: (y + 2)/4 − 3(y − 1) = 7/8

Question 3: In a class of 120 students, 1/4​ of the students do not participate in sports. How many students participate in sports?

Question 4: If a recipe calls for 2/3​ cup of butter, and you want to make three times the quantity, how much butter do you need?

Question 5: Sophie ordered a large cake and cut it into 10 equal slices. She ate 4/10 of the cake, and her friend Lily ate 3/10. How much of the cake is left?

Question 6: A box contains 150 marbles. Jack gives 1/3​ of the marbles to his cousin. How many marbles did Jack give to his cousin?

Question 7: Factor the expression and simplify the fraction: x² + 7x + 12/x² + 10x +24

Question 8: Factor the numerator and denominator, then simplify: x² − 16/x² + 4x + 3

Answer Key:

1. x ≈ 20.8 or x ≈−2.8x
2. y ≈ 4.7y or y ≈ −2.2y
3. 90 students
4. 2 cups of butter
5. 3/10​ of the cake
6. 50 candies
7. x + 3/x + 6
8. (x − 4)(x + 4)/(x + 3)(x + 1)