explain how to add or subtract fractions with different denominators, and why finding a common denominator…

1. Adding or subtracting fractions with different denominators:

Brief Explanations:

To add or subtract fractions with different denominators, first find the least - common denominator (LCD) of the fractions. The LCD is the least common multiple of the denominators. Then, rewrite each fraction as an equivalent fraction with the LCD as the denominator. After that, add or subtract the numerators and keep the common denominator. Simplify the resulting fraction if possible. For example, if adding $\frac{a}{b}$ and $\frac{c}{d}$, find the LCD of $b$ and $d$, say $L$. Rewrite $\frac{a}{b}=\frac{a\times\frac{L}{b}}{L}$ and $\frac{c}{d}=\frac{c\times\frac{L}{d}}{L}$, then add the numerators: $\frac{a\times\frac{L}{b}+c\times\frac{L}{d}}{L}$. Finding a common denominator is important because fractions represent parts of a whole. When the denominators are different, the 'wholes' are divided into different - sized parts. By finding a common denominator, we are essentially re - dividing the wholes into the same - sized parts, so that we can add or subtract the parts (numerators) directly.

Answer:

First find the least - common denominator (LCD) of the fractions, rewrite each fraction with the LCD as the denominator, then add or subtract the numerators and keep the common denominator. Simplify if possible. Finding a common denominator is important because it makes the parts (sub - divisions of the whole) of the fractions the same size, allowing for addition or subtraction of the numerators.

2. How the author's word choice communicates the theme:

Brief Explanations:

The author's word choice can evoke emotions, create imagery, and set a tone. Words with positive or negative connotations can influence how the reader perceives the theme. Descriptive words can paint a vivid picture related to the theme. For example, using words like "gloomy" and "desolate" can communicate a theme of despair, while "bright" and "hopeful" words can suggest a theme of optimism.

Answer:

The author's word choice evokes emotions, creates imagery, and sets a tone that helps convey the theme to the reader, through the use of connotations and descriptive language.

3. How geography influenced the growth and success of ancient civilizations:

Brief Explanations:

Geography provided natural resources like water (rivers such as the Nile in Egypt and the Tigris - Euphrates in Mesopotamia) for irrigation, drinking, and transportation. Fertile soil near rivers allowed for agriculture, which supported large populations. Natural barriers like deserts or mountains could protect the civilization from invasions. Access to trade routes via waterways or land routes also contributed to economic growth.

Answer:

Geography provided natural resources (water, fertile soil), natural barriers for protection, and access to trade routes, which all contributed to the growth and success of ancient civilizations like Egypt and Mesopotamia.