Ratio, Proportion, Indices, LogarithmPYQ Nov 18Question 789 of 211
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If x:y=z:7=4:11\displaystyle x:y=z:7=4:11 then x+y+zz\displaystyle \frac{x+y+z}{z} is

Options

A2\displaystyle 2
B3\displaystyle 3
C4\displaystyle 4
D5\displaystyle 5
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Correct Answer

✅ Option a — 2\displaystyle 2

All Options:

  • A2\displaystyle 2
  • B3\displaystyle 3
  • C4\displaystyle 4
  • D5\displaystyle 5

Detailed Solution & Explanation

• We are given the ratios x:y=z:7=4:11\displaystyle x:y=z:7=4:11. This means that all these ratios are equal to a common constant. • We can write this as: xy=411\displaystyle \frac{x}{y} = \frac{4}{11} (Equation 1) z7=411\displaystyle \frac{z}{7} = \frac{4}{11} (Equation 2) • From Equation 2, we can find the value of z\displaystyle z: z=7×411\displaystyle z = 7 \times \frac{4}{11} z=2811\displaystyle z = \frac{28}{11} • From Equation 1, we can express x\displaystyle x in terms of y\displaystyle y: x=y×411\displaystyle x = y \times \frac{4}{11} x=4y11\displaystyle x = \frac{4y}{11} • Now, we need to find the value of x+y+zz\displaystyle \frac{x+y+z}{z}. Let's substitute the expressions for x\displaystyle x and z\displaystyle z into this fraction: x+y+zz=4y11+y+28112811\displaystyle \frac{x+y+z}{z} = \frac{\frac{4y}{11} + y + \frac{28}{11}}{\frac{28}{11}} • To simplify the numerator, find a common denominator: 4y11+y+2811=4y11+11y11+2811=4y+11y+2811=15y+2811\displaystyle \frac{4y}{11} + y + \frac{28}{11} = \frac{4y}{11} + \frac{11y}{11} + \frac{28}{11} = \frac{4y+11y+28}{11} = \frac{15y+28}{11} • Now substitute this back into the main fraction: 15y+28112811\displaystyle \frac{\frac{15y+28}{11}}{\frac{28}{11}} • We can cancel out the 11\displaystyle 11 in the denominator of both the numerator and the denominator: 15y+2828\displaystyle \frac{15y+28}{28} • This expression still contains y\displaystyle y. Let's re-evaluate the initial ratios. A simpler approach is to use a common constant k\displaystyle k. • Let xy=z7=411=k\displaystyle \frac{x}{y} = \frac{z}{7} = \frac{4}{11} = k. • This implies: x=4k\displaystyle x = 4k y=11k\displaystyle y = 11k (from x:y=4:11  ⟹  xy=411\displaystyle x:y = 4:11 \implies \frac{x}{y} = \frac{4}{11}. If x=4k\displaystyle x=4k, then y=11k\displaystyle y=11k) z=7×411=2811\displaystyle z = 7 \times \frac{4}{11} = \frac{28}{11} (This is incorrect if we use k\displaystyle k for x\displaystyle x and y\displaystyle y. Let's restart with a consistent k\displaystyle k.) • Let x4=y11=z7=k\displaystyle \frac{x}{4} = \frac{y}{11} = \frac{z}{7} = k. This is the correct way to set up the ratios if x:y=4:11\displaystyle x:y=4:11 and z:7=4:11\displaystyle z:7=4:11. • From x:y=4:11\displaystyle x:y=4:11, we have x=4k1\displaystyle x=4k_1 and y=11k1\displaystyle y=11k_1. • From z:7=4:11\displaystyle z:7=4:11, we have z7=411\displaystyle \frac{z}{7} = \frac{4}{11}, so z=2811\displaystyle z = \frac{28}{11}. • This means the ratios are not directly x:y:z=4:11:7\displaystyle x:y:z = 4:11:7 in a simple way. • Let's use the given information directly: xy=411\displaystyle \frac{x}{y} = \frac{4}{11} z7=411\displaystyle \frac{z}{7} = \frac{4}{11} • From z7=411\displaystyle \frac{z}{7} = \frac{4}{11}, we get z=2811\displaystyle z = \frac{28}{11}. • We need to find x+y+zz\displaystyle \frac{x+y+z}{z}. • We can rewrite this as xz+yz+zz=xz+yz+1\displaystyle \frac{x}{z} + \frac{y}{z} + \frac{z}{z} = \frac{x}{z} + \frac{y}{z} + 1. • Let's find xz\displaystyle \frac{x}{z} and yz\displaystyle \frac{y}{z}. • We know xy=411\displaystyle \frac{x}{y} = \frac{4}{11}, so x=411y\displaystyle x = \frac{4}{11}y. • We know z=2811\displaystyle z = \frac{28}{11}. • Substitute x\displaystyle x and z\displaystyle z into xz\displaystyle \frac{x}{z}: xz=411y2811=4y28=y7\displaystyle \frac{x}{z} = \frac{\frac{4}{11}y}{\frac{28}{11}} = \frac{4y}{28} = \frac{y}{7}. • Now substitute z\displaystyle z into yz\displaystyle \frac{y}{z}: yz=y2811=11y28\displaystyle \frac{y}{z} = \frac{y}{\frac{28}{11}} = \frac{11y}{28}. • Now substitute these back into xz+yz+1\displaystyle \frac{x}{z} + \frac{y}{z} + 1: y7+11y28+1\displaystyle \frac{y}{7} + \frac{11y}{28} + 1 • Find a common denominator for the terms with y\displaystyle y: 4y28+11y28+1=4y+11y28+1=15y28+1\displaystyle \frac{4y}{28} + \frac{11y}{28} + 1 = \frac{4y+11y}{28} + 1 = \frac{15y}{28} + 1. • This still has y\displaystyle y. There must be a simpler way. • Let's use the property of ratios. If ab=cd=k\displaystyle \frac{a}{b} = \frac{c}{d} = k, then a=bk\displaystyle a=bk and c=dk\displaystyle c=dk. • We have xy=411\displaystyle \frac{x}{y} = \frac{4}{11}. This means x=4k\displaystyle x = 4k and y=11k\displaystyle y = 11k for some constant k\displaystyle k. • We also have z7=411\displaystyle \frac{z}{7} = \frac{4}{11}. This means z=7×411=2811\displaystyle z = 7 \times \frac{4}{11} = \frac{28}{11}. • Now substitute x\displaystyle x, y\displaystyle y, and z\displaystyle z into the expression x+y+zz\displaystyle \frac{x+y+z}{z}: 4k+11k+28112811\displaystyle \frac{4k + 11k + \frac{28}{11}}{\frac{28}{11}} =15k+28112811\displaystyle = \frac{15k + \frac{28}{11}}{\frac{28}{11}} $= \frac{15k}{\frac{28}{11}} + \

About This Chapter: Ratio, Proportion, Indices, Logarithm

Paper

Paper 3: Quantitative Aptitude

Weightage

5-7 Marks

Key Topics

Ratio, Proportion, Indices, Logarithms

This chapter covers Ratio, Proportion, Indices, Logarithms and is part of Paper 3: Quantitative Aptitude in the CA Foundation exam.

View Official ICAI Syllabus

Exam Strategy Tip

This topic carries 5-7 Marks weightage. Focus on understanding core concepts rather than memorizing.

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