Question : Three numbers are in the ratio $\frac{1}{2}: \frac{2}{3}: \frac{3}{4}$. If the difference between the greatest number and the smallest number is 33, then the HCF of the three numbers is:
Option 1: 9
Option 2: 5
Option 3: 13
Option 4: 11
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Correct Answer: 11
Solution : Let the three numbers as $x$, $y$, and $z$. Given that the numbers are in the ratio $⇒\frac{1}{2} : \frac{2}{3} : \frac{3}{4}$. $x : y : z = \frac{1}{2} : \frac{2}{3} : \frac{3}{4}$ Multiply all the ratios by the least common multiple (LCM) of 2, 3, and 4, which is 12. $⇒x : y : z = 6 : 8 : 9$ So, the numbers are $6k$, $8k$, and $9k$ for some integer $k$. Given that the difference between the greatest number and the smallest number is 33. $⇒9k - 6k = 33$ $⇒k = 11$ Therefore, the three numbers are $6k = 66$, $8k = 88$, and $9k = 99$. The highest common factor (HCF) of these three numbers is 11. Hence, the correct answer is 11.
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Question : Three numbers are in the ratio $\frac{4}{5}: \frac{5}{6}: \frac{9}{10}$. The difference between the smallest and the greatest numbers is 12. Find the number which is neither the smallest nor the greatest.
Question : Which of the following is the smallest ratio? $\frac{5}{6}, \frac{7}{9}, \frac{11}{12}, \frac{13}{18}$
Question : Choose the option in which the numbers are in the correct ascending order.
Question : Let $x=\frac{5 \frac{3}{4}-\frac{3}{7} \times 15 \frac{3}{4}+2 \frac{2}{35} \div 1 \frac{11}{25}}{\frac{3}{4} \div 5 \frac{1}{4}+5 \frac{3}{5} \div 3 \frac{4}{15}}$. When $y$ is added to $x$, the result is $\frac{7}{13}$. What is the value of $y$?
Question : If $(x-\frac{1}{3})^2+(y-4)^2=0$, then what is the value of $\frac{y+x}{y-x}$?
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