Comprehensive Review of Grade Level Mathematics: Number Theory, Fractions, and Algebra

Identification of Even Numbers and Divisibility Rules

In mathematics, an even number (referred to as "Tegamash") is a whole number that is exactly divisible by $2$. This means when the number is divided by $2$, the remainder is $0$. A simple rule to identify even numbers is to check the last digit: if the number ends in $0$, $2$, $4$, $6$, or $8$, it is even. In the provided set of whole numbers, the options include $3243$, $4789$, $8075$, and $4786$. Among these, $4786$ is the only number ending in an even digit ($6$), making it the even number.

Divisibility rules allow for the determination of whether a number can be divided by another without performing full long division. For a number to be divisible by $6$, it must simultaneously satisfy the divisibility rules for both $2$ and $3$. This means the number must be even and the sum of its digits must be divisible by $3$. Examining the options provided:

• $8156$: Ends in $6$ (divisible by $2$), but the sum of digits ($8+1+5+6 = 20$) is not divisible by $3$.
• $3627$: Ends in $7$ (not divisible by $2$).
• $7254$: Ends in $4$ (divisible by $2$). The sum of its digits ($7+2+5+4 = 18$) is divisible by $3$ ($18 \div 3 = 6$). Therefore, $7254$ is divisible by $6$.
• $9670$: Ends in $0$ (divisible by $2$), but the sum of digits ($9+6+7+0 = 22$) is not divisible by $3$.

Factors and Prime Factorization

A factor (or "Akfay") is a number that divides another number completely without leaving a remainder. In the exercise, several statements about factors were evaluated to find the correct relationship:

• $12$ is a factor of $128$: Incorrect, as $128 \div 12 = 10$ with a remainder of $8$.
• $12$ is a factor of $132$: Correct, because $132 \div 12 = 11$ with no remainder.
• $4$ is a factor of $154$: Incorrect, as $154 \div 4 = 38.5$.
• $9$ is a factor of $143$: Incorrect, as the sum of digits ($1+4+3=8$) is not divisible by $9$.

Prime factorization (referred to as "Bichenya Tinten") involves expressing a composite number as a product of its prime factors. For the number $240$, the procedure begins by dividing by the smallest prime numbers. The breakdown is as follows:

• $240 = 2 \times 120$
• $120 = 2 \times 60$
• $60 = 2 \times 30$
• $30 = 2 \times 15$
• $15 = 3 \times 5$ Combining these results, the prime factorization of $240$ is expressed as $2^4 \times 3 \times 5$.

Greatest Common Factor (GCF) and Sequence Applications

The Greatest Common Factor (GCF), or "Tilqu Ye-gara Akfay," is the largest positive integer that divides each of the integers. To find the GCF of $120$ and $84$, we examine their prime factors:

• Prime factorization of $120 = 2^3 \times 3 \times 5$
• Prime factorization of $84 = 2^2 \times 3 \times 7$
• The common prime factors are $2^2$ and $3$.
• GCF = $2^2 \times 3 = 4 \times 3 = 12$.

In practical scenarios involving sequences, the first meeting point of two people counting from different starting points with different intervals can be solved by listed their values.

• Martha starts at $12$ and counts with a difference of $6$: $12, 18, 24, 30, 36, ...$
• Birtukan starts at $10$ and counts with a difference of $5$: $10, 15, 20, 25, 30, 35, ...$ Comparing the two sequences, the first common number they both reach is $30$.

Fractions, Decimals, and Percentages

Simplifying a fraction to its lowest terms (or "Ziqtenya Hisabawi Qal") requires dividing both the numerator and the denominator by their greatest common factor. For the fraction $\frac{112}{72}$, the GCF is $8$:

• $112 \div 8 = 14$
• $72 \div 8 = 9$
• Result: $\frac{14}{9}$.

Understanding the relationship between fractions, decimals, and percentages is essential for comparison.

• Testing for equivalence: $\frac{7}{4} = 1.75$ is true because $7 \div 4 = 1.75$. However, several other listed options such as $\frac{2}{3} = 3.75$ are mathematically incorrect.
• Converting fractions to percentages: To convert $\frac{23}{50}$ to a percent, multiply by $100$.
• Calculation: $\frac{23}{50} \times 100 = 23 \times 2 = 46\%$.

Ordering fractions and decimals requires converting them to a common format. For example, comparing $\frac{1}{3}$, $0.34$, $0.5$, $\frac{5}{6}$, and $0.8$ involves noting that $\frac{1}{3} \approx 0.333$. Therefore, the sequence $0.333 < 0.34 < 0.5 < 0.8 < 0.833 (\frac{5}{6})$ represents the correct ascending order.

Applied Arithmetic and Financial Calculations

Word problems often require applying percentage formulas to currency. If Almaz has $140$ Birr and spends $45\%$ on a book, the total amount spent is calculated as:

• $\text{Cost} = \frac{45}{100} \times 140$
• $\text{Cost} = 0.45 \times 140 = 63$ Almaz spent $63$ Birr on the book.

Operations involving mixed decimals and fractions involve converting to a single format. For the product $\frac{1}{4} \times 0.5$:

• Convert $0.5$ to the fraction $\frac{1}{2}$.
• $\frac{1}{4} \times \frac{1}{2} = \frac{1}{8}$.

For complex verification tasks like checking equality in $4.2 \times 5.4 = 22.68$, multiplying out the decimals ($42 \times 54 = 2268$ and placing the decimal two places in) confirms the statement is true.

Variation and Algebraic Solving

In mathematics, variables can represent changing quantities. The relationship between the speed of a car and the time taken to cover a fixed distance is known as inverse proportionality (or "In-retu'e wederegna"). As the speed increases, the time required decreases, and vice versa.

To solve for a missing variable in an algebraic sentence, such as $\frac{x}{10} - 0.5 = 2.50$, isolate the variable $x$:

1. Add $0.5$ to both sides of the equation: $\frac{x}{10} = 2.50 + 0.5 = 3.0$
2. Multiply both sides by $10$ to solve for $x$: $x = 3.0 \times 10 = 30$ Thus, the whole number that makes the sentence true is $30$.