Fraction and Mixed-Number Sums - ISEE Middle Level Mathematics Achievement

1. What is $\frac{1}{3} + \frac{1}{6}$?

   $\frac{1}{2}$. Using LCD of 6, convert $\frac{1}{3}$ to $\frac{2}{6}$ and add to $\frac{1}{6}$ for $\frac{3}{6}$, simplifying to $\frac{1}{2}$.

2. What is $\frac{2}{5} + \frac{1}{2}$?

   $\frac{9}{10}$. LCD of 10 converts $\frac{2}{5}$ to $\frac{4}{10}$ and $\frac{1}{2}$ to $\frac{5}{10}$, summing to $\frac{9}{10}$.

3. What is $\frac{11}{12} + \frac{1}{8}$?

   $1\ \frac{1}{24}$. LCD of 24: $\frac{22}{24}+\frac{3}{24}=\frac{25}{24}=1\frac{1}{24}$.

4. What is $\frac{5}{6} - \frac{1}{4}$?

   $\frac{7}{12}$. With LCD of 12, $\frac{5}{6}$ becomes $\frac{10}{12}$ minus $\frac{3}{12}$ equals $\frac{7}{12}$.

5. What is $2\ \frac{7}{8} - 1\ \frac{1}{4}$?

   $1\ \frac{5}{8}$. Convert $1\frac{1}{4}$ to $1\frac{2}{8}$; since $\frac{7}{8}>\frac{2}{8}$, subtract to get $1\frac{5}{8}$.

6. What is $\frac{7}{10} - \frac{3}{10}$?

   $\frac{2}{5}$. Subtracting numerators over the shared denominator of 10 gives $\frac{4}{10}$, reducing to $\frac{2}{5}$.

7. What is $3\ \frac{2}{3} + 1\ \frac{1}{6}$?

   $4\ \frac{5}{6}$. Convert to improper fractions $\frac{11}{3}+\frac{7}{6}$, use LCD 6 for $\frac{22}{6}+\frac{7}{6}=\frac{29}{6}=4\frac{5}{6}$.

8. What is the first step to add $\frac{a}{b}$ and $\frac{c}{d}$ when $b \ne d$?

   Rewrite both with a common denominator (often $\text{LCD}(b,d)$). To add fractions with unlike denominators, equivalent fractions must share a common denominator, ideally the least common one, before combining numerators.

9. What is $3\ \frac{1}{6} - \frac{5}{12}$?

   $2\ \frac{3}{4}$. $\frac{19}{6}-\frac{5}{12}$ with LCD 12: $\frac{38}{12}-\frac{5}{12}=\frac{33}{12}=\frac{11}{4}=2\frac{3}{4}$.

10. What is the key step when subtracting mixed numbers if the top fraction is smaller than the bottom fraction?

    Regroup: borrow $1$ from the whole number and add it to the fraction part. Borrowing converts 1 from the whole number into an equivalent fraction, allowing subtraction when the minuend fraction is smaller.

11. What is $1\ \frac{3}{10} + \frac{7}{10}$?

    $2$. Add fractions $\frac{3}{10}+\frac{7}{10}=\frac{10}{10}=1$, then plus whole 1 totals 2.

12. What is $1\ \frac{1}{4} + 2\ \frac{3}{4}$?

    $4$. Summing whole numbers and fractions separately: $1+2=3$ and $\frac{1}{4}+\frac{3}{4}=1$, totaling $4$.

13. What is $2\ \frac{1}{3} + \frac{5}{6}$?

    $3\ \frac{1}{6}$. As improper, $\frac{7}{3}+\frac{5}{6}$ with LCD 6: $\frac{14}{6}+\frac{5}{6}=\frac{19}{6}=3\frac{1}{6}$.

14. What is the improper fraction form of the mixed number $a\ \frac{b}{c}$?

    $\frac{ac+b}{c}$. Convert the mixed number by multiplying the whole part by the denominator and adding the numerator, placing over the original denominator.

15. What is the simplest method to add $\frac{a}{b} + \frac{c}{d}$ using a common denominator?

    $\frac{ad+bc}{bd}$ (then simplify if possible). Using $bd$ as the common denominator, convert each fraction and add numerators, then reduce the resulting fraction if possible.

16. What is the rule for subtracting fractions with the same denominator: $\frac{a}{n} - \frac{b}{n}$?

    $\frac{a-b}{n}$. For subtraction with matching denominators, subtract the numerators and keep the common denominator for the result.

17. What is the least common denominator (LCD) of fractions with denominators $m$ and $n$?

    $\text{LCD} = \text{LCM}(m,n)$. The least common denominator equals the least common multiple of the denominators, enabling equivalent fraction conversion for addition or subtraction.

18. What is $\frac{3}{8} + \frac{1}{8}$?

    $\frac{1}{2}$. Adding numerators over the common denominator of 8 yields $\frac{4}{8}$, which simplifies to $\frac{1}{2}$.

19. What is the rule for adding fractions with the same denominator: $\frac{a}{n} + \frac{b}{n}$?

    $\frac{a+b}{n}$. When denominators are identical, numerators are added directly while retaining the shared denominator to form the sum.

20. What is the simplest method to subtract $\frac{a}{b} - \frac{c}{d}$ using a common denominator?

    $\frac{ad-bc}{bd}$ (then simplify if possible). With $bd$ as the common denominator, subtract the converted numerators and simplify the outcome for the difference.

21. What is $5\ \frac{1}{2} - 2\ \frac{1}{4}$?

    $3\ \frac{1}{4}$. As improper fractions, $\frac{11}{2}-\frac{9}{4}$ with LCD 4: $\frac{22}{4}-\frac{9}{4}=\frac{13}{4}=3\frac{1}{4}$.

22. What is $\frac{9}{10} - \frac{2}{5}$?

    $\frac{1}{2}$. Convert $\frac{2}{5}$ to $\frac{4}{10}$; subtract from $\frac{9}{10}$ to get $\frac{5}{10}$, simplifying to $\frac{1}{2}$.

23. What should you do after adding or subtracting fractions to give a final answer?

    Simplify the fraction and write improper results as mixed numbers if needed. Reducing to lowest terms ensures the simplest form, and converting improper fractions to mixed numbers improves readability when appropriate.

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