26. a. Show that the ratios Start Fraction 10 over 20 End Fraction and Start Fraction 30 over 60 End Fraction form a proportion by finding a common multiplier.
b. Show that the ratios in part (a) are equal by writing them in simplest form.
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10/20
30/60
Divide both numerators and denominators by 10
1/2 = 3/6
3/6 and be simplified by dividing numerator and denominator by 3.
1/2 = 1/2
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To show that the ratios Start Fraction 10 over 20 End Fraction and Start Fraction 30 over 60 End Fraction form a proportion, we need to find a common multiplier.
a. Finding a common multiplier:
To find a common multiplier, we need to identify the lowest common denominator (LCD) of the fractions. In this case, both fractions have the same denominator of 20 and 60. Therefore, the LCD is 20.
To convert the first fraction Start Fraction 10 over 20 End Fraction to have a denominator of 20, we can multiply both the numerator and denominator by 2:
Start Fraction 10 over 20 End Fraction * (2/2) = Start Fraction 20 over 40 End Fraction
To convert the second fraction Start Fraction 30 over 60 End Fraction to have a denominator of 20, we can multiply both the numerator and denominator by 1/3:
Start Fraction 30 over 60 End Fraction * (1/3) = Start Fraction 10 over 20 End Fraction
Now, both fractions have a common denominator of 20: Start Fraction 20 over 40 End Fraction and Start Fraction 10 over 20 End Fraction.
Since the fractions are equivalent, they form a proportion.
b. To show that the ratios in part (a) are equal, we need to write them in simplest form:
Start Fraction 20 over 40 End Fraction simplifies to Start Fraction 1 over 2 End Fraction (by dividing both the numerator and denominator by 20).
Start Fraction 10 over 20 End Fraction is already in its simplest form, which is also Start Fraction 1 over 2 End Fraction.
Therefore, both ratios are equal when written in simplest form, showing that they are equivalent.