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Five-fourths of a number is greater than three-fourths of the number by 7. Find the number.
8
14
12
10
14
Since the denominators are the same, just subtract the numerators: (5−3)=2. So, 42x=7⟹21x=7, which means x=14.
Five-fourths of a number is greater than three-fourths of the same number by 7.
45x−43x=7
Since the denominators are the same, just subtract the numerators: (5−3)=2. So, 42x=7⟹21x=7, which means x=14.
Confusing 'greater than ... by 7' with 'is equal to 7', or making errors when subtracting fractions with different denominators (though these denominators are already matched).
Define the variable
Let the unknown number be x.
x
Set up the equation
Translate the phrase 'Five-fourths of a number is greater than three-fourths of the number by 7' into an algebraic equation.
45x−43x=7
Simplify and solve
Subtract the fractions: 45−3x=7, which simplifies to 42x=7. Reducing the fraction gives 21x=7. Multiplying both sides by 2 yields x=14.
21x=7⟹x=14
B is correct because the difference between five-fourths and three-fourths of the number is two-fourths (or one-half), and one-half of 14 is 7.
This concept of 'difference' is fundamental in arithmetic progression and speed-distance-time problems where you calculate the relative speed (difference) to find time or distance.
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