Olympiad Test: Rational Numbers - Class 7 MCQ

To find the product of ^-7/₈ and ⁴/₂₁, we multiply the numerators and the denominators:

Product = (-7/8) × (4/21) = (-7 × 4) / (8 × 21)

Now calculate:

• Numerator: (-7 × 4) = -28
• Denominator: (8 × 21) = 168

So, the product is:

(-28) / 168

Now, simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 28:

(-28 ÷ 28) / (168 ÷ 28) = -1 / 6

Thus, the product is ^-1/₆, and the correct answer is -1/6.

We are asked to find what number should be added to ⁷/₁₂ to get ⁴/₁₅. This can be written as:

To solve for x, subtract (7/12) from both sides:

Now, to subtract these fractions, we need a common denominator. The least common denominator (LCD) of 15 and 12 is 60.

Convert both fractions to have a denominator of 60:

Now, subtract the fractions:

Thus, the number that should be added to ⁷/₁₂ to get ⁴/₁₅ is ^-19/₆₀, and the correct answer is -19/60.

To simplify the fraction ^-44/₇₂, we need to find the greatest common divisor (GCD) of 44 and 72 and divide both the numerator and the denominator by that GCD.

(-44) / 72 = (-44 ÷ 4) / (72 ÷ 4) = -11 / 18

1. Find the GCD of 44 and 72:
   • The factors of 44 are 1, 2, 4, 11, 22, and 44.
   • The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72.
   • The greatest common divisor is 4.
2. Now, divide both the numerator and the denominator by 4:
   

Thus, the simplified form of ^-44/₇₂ is ^-11/₁₈, and the correct answer is -11/18.

We are asked to find what should be added to ^-5/₁₁ to get ³/₁₁. This can be written as:

(-5/11) + x = (3/11)

To solve for x, add ⁵/₁₁ to both sides:

x = (3/11) + (5/11)

Since the denominators are the same, we can add the numerators directly:

x = (3 + 5) / 11 = 8/11

Thus, the number that should be added to ^-5/₁₁ to get ³/₁₁ is ⁸/₁₁, and the correct answer is 8/11.

A rational number is positive if its numerator and denominator both have the same sign (either both positive or both negative). Let's analyze the options:

• 3/7: Both 3 and 7 are positive, so 3/7​ is positive.
• -5/7: The numerator is negative and the denominator is positive, so −5/7 is negative.
• -4/7: The numerator is negative and the denominator is positive, so −4/7​ is negative.
• -3/7: The numerator is negative and the denominator is positive, so −3/7​ is negative.

Thus, the only positive rational number is 3/7.

Step 1: Calculate the Numerator

The numerator is given as: 4 × (-7)

4 × (-7) = -28

So, the numerator is -28.

Step 2: Calculate the Denominator

The denominator is given as: (3 - 7) × (15 - 11)

First, calculate 3 - 7:

3 - 7 = -4

Next, calculate 15 - 11:

15 - 11 = 4

Now, multiply these two results:

(-4) × 4 = -16

So, the denominator is -16.

Step 3: Write the Rational Number

Now we can write the rational number as:

-28 / -16

Step 4: Simplify the Rational Number

Since both the numerator and denominator are negative, we can cancel the negative signs:

-28 / -16 = 28 / 16

Now, simplify 28/16:

Both 28 and 16 can be divided by 4:

28 ÷ 4 = 7, 16 ÷ 4 = 4

The rational number is: 7/4

• If we talk about 1/3 = 0.333 and 1/4 = 0.25 
• Obviously , 1/3 is greater than 1/4 
• But as in the question it is given the comparison between -1/3 and -1/4
• -1/4 is greater than -1/3

Hence, -1/3 < -1/4

0 is greater than any negative number.

-7/6 is a negative number, so 0 is greater than -7/6.

Therefore, the correct symbol to fill in the blank is <, since 0 is less than -7/6 when placed in the order.

The correct symbol is <, and the answer is Option B: <.

We are asked to fill in the blank with the appropriate symbol to compare 5/-11 and -5/11.

5/-11 is equivalent to -5/11 because the negative sign is in the denominator.

-5/11 is a negative number, and comparing both fractions, we see that they are equal.

The correct symbol is =, and the answer is Option C: =.

We are asked to find what number should be added to ³/₈ to get ^-1/₂₄. This can be written as:

(3/8) + x = (-1/24)

To solve for x, subtract ³/₈ from both sides:

x = (-1/24) - (3/8)

Now, to subtract these fractions, we need a common denominator. The least common denominator (LCD) of 24 and 8 is 24.

Convert both fractions to have a denominator of 24:

(3/8) = (3 * 3)/(8 * 3) = 9/24

Now, subtract the fractions:

x = (-1/24) - (9/24) = (-1 - 9)/24 = -10/24

Simplify the fraction:

-10/24 = -5/12

The number that should be added to ³/₈ to get ^-1/₂₄ is ^-5/₁₂, and the correct answer is Option A: -5/12.

Combine the fractions with the same denominator first.
For the fractions with denominator 4:

Combine the fractions with denominator 5:

Combine the fractions with denominator 3:

Now, the expression becomes:

Simplify the expression:

Convert -3 and 2 into fractions with denominator 2:

So the expression becomes:

Thus, the final answer is 3/2.

Rahul walks 6/7​ km towards his school and then returns 5/6​ km towards his home. To find how far he is from his home, we subtract the distance he has traveled back from the distance he initially covered.

So, the distance Rahul is from his home is:

To subtract these fractions, we need to find a common denominator. The least common denominator (LCD) of 7 and 6 is 42.

Now, convert both fractions to have a denominator of 42:

Now, subtract the fractions:

Thus, Rahul is 1/42​ km from his home.

Final Answer: Option A: 1/42 km.

First, calculate the product of the fractions:

x = (-1/3) × (-15/6)

Multiply the numerators and the denominators:

x = (-1 * -15) / (3 * 6) = 15 / 18

Now, simplify the fraction:

15 / 18 = 5 / 6

Next, find the reciprocal of 5/6:

Reciprocal = 6 / 5

The reciprocal is 6/5, so the correct answer is Option B: 6/5.

We are given that the product of two rational numbers is -8/9, and one of the numbers is -10/3. We need to find the other number.

Let the unknown number be x. From the information, we know:

(-10/3) × x = -8/9

To solve for x, divide both sides of the equation by -10/3:

x = (-8/9) ÷ (-10/3)

When dividing two fractions, we multiply by the reciprocal of the second fraction:

x = (-8/9) × (3/-10)

Simplifying the expression:

x = (-8 × 3) / (9 × -10) = -24 / -90

Since both the numerator and denominator are negative, we can cancel the negative signs:

x = 24 / 90

Now, simplify the fraction by dividing both the numerator and denominator by their greatest common divisor (GCD), which is 6:

x = (24 ÷ 6) / (90 ÷ 6) = 4/15

We are asked to reduce −63/99​ to its standard form. To do this, we need to simplify the fraction by finding the greatest common divisor (GCD) of the numerator and the denominator.

1. Find the GCD of 63 and 99:

   • The factors of 63 are: 1, 3, 7, 9, 21, 63.
   • The factors of 99 are: 1, 3, 9, 11, 33, 99.
   • The GCD of 63 and 99 is 9.

2. Divide both the numerator and the denominator by the GCD (9):
   

Thus, the simplified fraction is −7/11​, which is the standard form.

Answer: Option B: -7/11.

• The smallest 2-digit number is 10.
• The greatest 3-digit number is 999.

Thus, the rational number with the smallest 2-digit number as the denominator and the greatest 3-digit number as the numerator is:
999/10
Answer: Option B: 999/10.

The rational number 9/1​ is simply 999 because any number divided by 1 remains the same.

Thus, the integer equivalent of 9/1​ is 9.

Answer: Option B: 9.

The reciprocal of a number is obtained by flipping the numerator and denominator.

1. The reciprocal of 4 is 1/4.
2. The reciprocal of 1/4 is 4.

Thus, the reciprocal of the reciprocal of 4 is 4.

The correct answer is Option D: 4.

We are asked to find two rational numbers between -3/4 and 0.

Let's list a few rational numbers between -3/4 and 0:

• -3/4 is slightly less than -1, and 0 is the reference for a positive number.
• Rational numbers between them could be fractions with a smaller absolute value, such as -2/4 and -1/4.

Therefore, two rational numbers between -3/4 and 0 are -2/4 and -1/4.

The correct answer is Option C: -2/4 and -1/4.

The reciprocal of a number is obtained by flipping the numerator and denominator. For any non-zero number x, the reciprocal is 1/x.

In the case of -1, the reciprocal of -1 is:

1 / -1 = -1

Thus, the reciprocal of -1 is -1.