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Which number, when added to each of the numbers 3, 5, 13, and 19, will make the resulting numbers in proportion?
1
2
3
4
2
Test the options starting from the smallest. If x=1, then 4/6 = 2/3 and 14/20 = 7/10. Wait, 4/6=2/3 and 14/20=7/10. Let's re-verify: (3+1)/(5+1) = 4/6 = 2/3. (13+1)/(19+1) = 14/20 = 7/10. Option A is 1. Checking x=2: (3+2)/(5+2) = 5/7 and (13+2)/(19+2) = 15/21 = 5/7. Thus, 2 is the correct answer.
The numbers are 3, 5, 13, and 19. Let the added number be x such that (3+x)/(5+x) = (13+x)/(19+x).
b+xa+xтАЛ=d+xc+xтАЛ
Test the options starting from the smallest. If x=1, then 4/6 = 2/3 and 14/20 = 7/10. Wait, 4/6=2/3 and 14/20=7/10. Let's re-verify: (3+1)/(5+1) = 4/6 = 2/3. (13+1)/(19+1) = 14/20 = 7/10. Option A is 1. Checking x=2: (3+2)/(5+2) = 5/7 and (13+2)/(19+2) = 15/21 = 5/7. Thus, 2 is the correct answer.
Many students mistakenly assume the ratio must be equal to 1, or they solve the quadratic equation incorrectly by missing a sign during cross-multiplication.
Set up the proportion equation
To make 3, 5, 13, 19 in proportion by adding x, we set up the equation: (3+x):(5+x)=(13+x):(19+x)
5+x3+xтАЛ=19+x13+xтАЛ
Cross-multiply the terms
Perform cross-multiplication: (3+x)(19+x)=(5+x)(13+x)
57+3x+19x+x2=65+5x+13x+x2
Simplify and solve for x
Subtract x2 from both sides and combine the linear terms: 57+22x=65+18x. Then, isolate x: 22xтИТ18x=65тИТ57
4x=8тЯ╣x=2
B is correct because solving the equation 5+x3+xтАЛ=19+x13+xтАЛ yields x=2.
This type of problem appears frequently in variations involving subtraction or specific ratios, which can also be solved using the property of componendo and dividendo.
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