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Show why these fractions are equivalent: a) 4/28 = 1/7 b) 24/60 = 2/5 c) 25/125 = 1/5

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Here are the solutions for the problems:

1. Show why these fractions are equivalent.

a) 428=17\frac{4}{28} = \frac{1}{7}

  1. Find the factors of the numerator and the denominator.

    Factors of 4: {1,2,4}\{1, 2, 4\}
    Factors of 28: {1,2,4,7,14,28}\{1, 2, 4, 7, 14, 28\}

  2. Identify the Highest Common Factor (HCF).

    The HCF of 4 and 28 is 4.

  3. Divide both the numerator and the denominator by the HCF.

    4÷428÷4=17\frac{4 \div 4}{28 \div 4} = \frac{1}{7}

    Thus, 428\frac{4}{28} is equivalent to 17\frac{1}{7}.

    b) 2460=25\frac{24}{60} = \frac{2}{5}

  4. Find the factors of the numerator and the denominator.

    Factors of 24: {1,2,3,4,6,8,12,24}\{1, 2, 3, 4, 6, 8, 12, 24\}
    Factors of 60: {1,2,3,4,5,6,10,12,15,20,30,60}\{1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60\}

  5. Identify the Highest Common Factor (HCF).

    The HCF of 24 and 60 is 12.

  6. Divide both the numerator and the denominator by the HCF.

    24÷1260÷12=25\frac{24 \div 12}{60 \div 12} = \frac{2}{5}

    Thus, 2460\frac{24}{60} is equivalent to 25\frac{2}{5}.

    c) 25125=15\frac{25}{125} = \frac{1}{5}

  7. Find the factors of the numerator and the denominator.

    Factors of 25: {1,5,25}\{1, 5, 25\}
    Factors of 125: {1,5,25,125}\{1, 5, 25, 125\}

  8. Identify the Highest Common Factor (HCF).

    The HCF of 25 and 125 is 25.

  9. Divide both the numerator and the denominator by the HCF.

    25÷25125÷25=15\frac{25 \div 25}{125 \div 25} = \frac{1}{5}

    Thus, 25125\frac{25}{125} is equivalent to 15\frac{1}{5}.

    2. Calculate and simplify fractions that are multiples of each other.

    a) 24+78−16\frac{2}{4} + \frac{7}{8} - \frac{1}{6}

  10. Find the Least Common Multiple (LCM) of the denominators 4, 8, and 6.

    Multiples of 4: 4, 8, 12, 16, 20, 24, ...
    Multiples of 8: 8, 16, 24, ...
    Multiples of 6: 6, 12, 18, 24, ...
    The LCM of 4, 8, and 6 is 24.

  11. Convert each fraction to an equivalent fraction with a denominator of 24.

    24=2×64×6=1224\frac{2}{4} = \frac{2 \times 6}{4 \times 6} = \frac{12}{24} 78=7×38×3=2124\frac{7}{8} = \frac{7 \times 3}{8 \times 3} = \frac{21}{24} 16=1×46×4=424\frac{1}{6} = \frac{1 \times 4}{6 \times 4} = \frac{4}{24}

  12. Perform the addition and subtraction.

    1224+2124−424=12+21−424=33−424=2924\frac{12}{24} + \frac{21}{24} - \frac{4}{24} = \frac{12 + 21 - 4}{24} = \frac{33 - 4}{24} = \frac{29}{24}

  13. Convert the improper fraction to a mixed number.

    2924=1524\frac{29}{24} = 1 \frac{5}{24}

    The simplified answer is 1524\boxed{1 \frac{5}{24}}.

    b) 910−35−28+78\frac{9}{10} - \frac{3}{5} - \frac{2}{8} + \frac{7}{8}

  14. Find the Least Common Multiple (LCM) of the denominators 10, 5, and 8.

    Multiples of 10: 10, 20, 30, 40, ...
    Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, ...
    Multiples of 8: 8, 16, 24, 32, 40, ...
    The LCM of 10, 5, and 8 is 40.

  15. Convert each fraction to an equivalent fraction with a denominator of 40.

    910=9×410×4=3640\frac{9}{10} = \frac{9 \times 4}{10 \times 4} = \frac{36}{40} 35=3×85×8=2440\frac{3}{5} = \frac{3 \times 8}{5 \times 8} = \frac{24}{40} 28=2×58×5=1040\frac{2}{8} = \frac{2 \times 5}{8 \times 5} = \frac{10}{40} 78=7×58×5=3540\frac{7}{8} = \frac{7 \times 5}{8 \times 5} = \frac{35}{40}

  16. Perform the addition and subtraction.

    3640−2440−1040+3540=36−24−10+3540\frac{36}{40} - \frac{24}{40} - \frac{10}{40} + \frac{35}{40} = \frac{36 - 24 - 10 + 35}{40} =12−10+3540= \frac{12 - 10 + 35}{40} =2+3540= \frac{2 + 35}{40} =3740= \frac{37}{40}

    The simplified answer is 3740\boxed{\frac{37}{40}}.

    c) 26+512+112\frac{2}{6} + \frac{5}{12} + \frac{1}{12}

  17. Find the Least Common Multiple (LCM) of the denominators 6 and 12.

    Multiples of 6: 6, 12, ...
    Multiples of 12: 12, ...
    The LCM of 6 and 12 is 12.

  18. Convert the fraction 26\frac{2}{6} to an equivalent fraction with a denominator of 12.

    26=2×26×2=412\frac{2}{6} = \frac{2 \times 2}{6 \times 2} = \frac{4}{12}

    The other fractions already have a denominator of 12.

  19. Perform the addition.

    412+512+112=4+5+112=1012\frac{4}{12} + \frac{5}{12} + \frac{1}{12} = \frac{4 + 5 + 1}{12} = \frac{10}{12}

  20. Simplify the fraction by dividing the numerator and denominator by their HCF.

    The HCF of 10 and 12 is 2.

    10÷212÷2=56\frac{10 \div 2}{12 \div 2} = \frac{5}{6}

    The simplified answer is 56\boxed{\frac{5}{6}}.

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