Math worksheet with five questions on fractions, reciprocals, and number lines.

Let's solve each question step by step from the Question Bank you provided.

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1. Find using distributivity:

$$
\left(\frac{9}{16} \times \frac{4}{12}\right) + \left(\frac{9}{16} \times \frac{-3}{9}\right)
$$

We can use the distributive property:
$$
a \times b + a \times c = a \times (b + c)
$$

Here, $ a = \frac{9}{16} $, $ b = \frac{4}{12} $, $ c = \frac{-3}{9} $

So,
$$
\frac{9}{16} \times \left( \frac{4}{12} + \frac{-3}{9} \right)
$$

Now simplify the terms inside the parentheses:

- $ \frac{4}{12} = \frac{1}{3} $
- $ \frac{-3}{9} = -\frac{1}{3} $

So:
$$
\frac{1}{3} + \left(-\frac{1}{3}\right) = 0
$$

Now:
$$
\frac{9}{16} \times 0 = 0
$$

✔ Answer: 0

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2. Using appropriate properties, find:

$$
\frac{2}{5} \times \left( \frac{-3}{7} \right) - \frac{1}{6} \cdot \frac{3}{2} + \frac{1}{4} \times \frac{2}{5}
$$

Let’s simplify term by term.

#### First term:
$$
\frac{2}{5} \times \left( \frac{-3}{7} \right) = \frac{-6}{35}
$$

#### Second term:
$$
\frac{1}{6} \times \frac{3}{2} = \frac{3}{12} = \frac{1}{4}
$$

Note: The expression has a minus sign before this term, so:
$$
- \frac{1}{4}
$$

#### Third term:
$$
\frac{1}{4} \times \frac{2}{5} = \frac{2}{20} = \frac{1}{10}
$$

Now combine all:
$$
\frac{-6}{35} - \frac{1}{4} + \frac{1}{10}
$$

Find a common denominator. LCM of 35, 4, 10 is 140.

Convert each fraction:

- $ \frac{-6}{35} = \frac{-6 \times 4}{140} = \frac{-24}{140} $
- $ \frac{-1}{4} = \frac{-35}{140} $
- $ \frac{1}{10} = \frac{14}{140} $

Add:
$$
\frac{-24 - 35 + 14}{140} = \frac{-45}{140}
$$

Simplify:
$$
\frac{-45}{140} = \frac{-9}{28} \quad \text{(divided numerator and denominator by 5)}
$$

✔ Answer: $ \frac{-9}{28} $

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3. Write:

#### a. The rational number that does not have a reciprocal.

> A rational number does not have a reciprocal if its reciprocal would involve division by zero.

The only rational number like this is 0, because $ \frac{1}{0} $ is undefined.

✔ Answer: 0

#### b. The rational numbers that are equal to their reciprocals.

Let $ x = \frac{1}{x} $

Then:
$$
x^2 = 1 \Rightarrow x = \pm 1
$$

Check:
- $ 1 \times 1 = 1 $ → reciprocal is 1 ✔
- $ -1 \times -1 = 1 $ → reciprocal is -1 ✔

✔ Answer: 1 and -1

#### c. The rational number that is equal to its negative.

Let $ x = -x $

Then:
$$
2x = 0 \Rightarrow x = 0
$$

✔ Answer: 0

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4. Fill in the blanks:

#### a. Zero has ________ reciprocal.

Zero has no reciprocal.

✔ Answer: no

#### b. The reciprocal of $-5$ is ________.

Reciprocal of $-5$ is $ \frac{1}{-5} = -\frac{1}{5} $

✔ Answer: $-\frac{1}{5}$

#### c. The reciprocal of $ \frac{1}{x} $, where $ x \neq 0 $, is ________.

Reciprocal of $ \frac{1}{x} $ is $ x $

✔ Answer: $ x $

#### d. The product of two rational numbers is always a ________.

Product of two rational numbers is always a rational number.

✔ Answer: rational number

#### e. The reciprocal of a positive rational number is __________.

A positive rational number has a positive reciprocal.

Example: $ \frac{2}{3} $ → reciprocal $ \frac{3}{2} $ → positive

✔ Answer: positive rational number

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5. Represent these numbers on the number line:

#### a. $ \frac{7}{4} $

This is $ 1.75 $, which lies between 1 and 2, closer to 2.

On the number line:
- Mark 1 and 2.
- Divide into 4 parts: $ \frac{1}{4}, \frac{2}{4}, \frac{3}{4}, \frac{4}{4}=1 $
- $ \frac{7}{4} = 1 + \frac{3}{4} $ → So go 3/4 beyond 1.

✔ Place a point at 1.75

#### b. $ \frac{-5}{6} $

This is approximately $ -0.833 $, between -1 and 0, closer to -1.

On the number line:
- Mark -1 and 0.
- Divide into 6 parts: $ -\frac{1}{6}, -\frac{2}{6}, ..., -\frac{5}{6} $
- $ \frac{-5}{6} $ is just $ \frac{1}{6} $ short of -1.

✔ Place a point at about -0.833

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✔ Final Answers Summary:

1. 0
2. $ \frac{-9}{28} $
3.
a. 0
b. 1 and -1
c. 0
4.
a. no
b. $ -\frac{1}{5} $
c. $ x $
d. rational number
e. positive rational number
5.
a. $ \frac{7}{4} = 1.75 $ → mark between 1 and 2
b. $ \frac{-5}{6} \approx -0.833 $ → mark between -1 and 0

Let me know if you'd like a visual sketch of the number line!