A tea shop offers tea in cups of three different sizes. The product of the prices, in INR, of three different sizes is equal to 800. The prices of the smallest size and the medium size are in the ratio 2 : 5. If the shop owner decides to increase the prices of the smallest and the medium ones by INR 6 keeping the price of the largest size unchanged, the product then changes to 3200. The sum of the original prices of three different sizes, in INR, is
Solution and Explanation
Let the prices of the small, medium, and large cups be \(2x\), \(5x\), and \(y\) respectively.
Step 1: Form the equations
From the problem:
\((2x)(5x)(y) = 800 \quad \text{(1)}\)
After increasing the prices of the small and medium cups by 6, the product becomes:
\((2x + 6)(5x + 6)(y) = 3200 \quad \text{(2)}\)
Step 2: Eliminate \(y\)
Divide Equation (2) by Equation (1):
\(\frac{(2x + 6)(5x + 6)(y)}{(2x)(5x)(y)} = \frac{3200}{800}\)
Simplifies to:
\(\frac{(2x+6)(5x+6)}{10x^2} = 4\)
Step 3: Solve the equation
Multiply both sides by \(10x^2\):
\((2x+6)(5x+6) = 40x^2\)
Expand the left-hand side:
\(10x^2 + 30x + 12x + 36 = 10x^2 + 42x + 36\)
Rearranged:
\(10x^2 + 42x + 36 = 40x^2\)
\(30x^2 - 42x - 36 = 0\)
Divide by 2:
\(15x^2 - 21x - 18 = 0\)
Step 4: Factorize
\((5x + 6)(3x - 2) = 0\)
So, \(x = -\frac{6}{5}\) or \(x = 2\)
Since price can't be negative, take \(x = 2\)
Step 5: Find \(y\)
Substitute into Equation (1):
\((2 \cdot 2)(5 \cdot 2)(y) = 800\)
\(4 \cdot 10 \cdot y = 800 \Rightarrow 40y = 800 \Rightarrow y = 20\)
Final Answer:
\(2x + 5x + y = 4 + 10 + 20 = \boxed{34}\)
Top Questions on Ratio and Proportion
- In a business school, only three specializations are offered: marketing, finance, and operations. A student can opt for either one, or two, or no specializations. In the current batch of 120 students, 60 are specializing in marketing, 50 are specializing in finance and 30 are specializing in operations. During the placement process, all students specializing either in marketing or in finance are shortlisted for consulting job interviews.
- XAT - 2026
- Quantitative Ability and Data Interpretation
- Ratio and Proportion
A shopkeeper marks his goods 40% above cost price and offers a discount of 20%. What is his overall profit percentage?
- XAT - 2026
- General Aptitude
- Ratio and Proportion
Health insurance plays a vital role in ensuring financial protection and access to quality healthcare. In India, however, the extent and nature of health insurance coverage vary significantly between urban and rural areas. While urban populations often have better access to organized insurance schemes, employer-provided coverage, and awareness about health policies, rural populations face challenges such as limited outreach of insurance schemes, inadequate infrastructure, and lower awareness levels. This urban-rural divide in health insurance coverage highlights the broader issue of healthcare inequality, making it essential to analyze the factors contributing to this gap and explore strategies for more inclusive health protection. A state-level health survey was conducted.
The survey covered 1,80,000 adults across urban and rural areas. Urban residents formed 55% of the sample (that is, 99,000 people) while rural residents made up 45% (that is, 81,000 people). In each area, coverage was classified under four heads – Public schemes, Private insurance, Employer-provided coverage, and Uninsured. In urban areas, Public coverage accounted for 28% of the urban population, Private for 22%, Employer for 18%, and the remaining 32% were Uninsured. In rural areas, where formal coverage is generally lower, Public coverage stood at 35%, Private at 10%, Employer at 8%, while 47% were Uninsured.
For this survey, “Insured” includes everyone covered by Public + Private + Employer schemes, and “Uninsured” indicates those with no coverage at all. Officials noted that public schemes remain the backbone of rural coverage, while employer and private plans are relatively more prevalent in urban centres. (250 words)
- CLAT - 2026
- Quantitative Aptitude
- Ratio and Proportion
- If 36: 84 :: 42: X, then the value of X, is:
- CUET (UG) - 2025
- General Aptitude
- Ratio and Proportion
- Reema had 'n' chocolates. She distributed them among \(4\) children in the ratio of \(\frac{1}{2} : \frac{1}{3} : \frac{1}{5} : \frac{1}{8}\). If she gave them each one a complete chocolate, the minimum number of chocolates she had distributed.
- CMAT - 2025
- Quantitative Ability and Data Interpretation
- Ratio and Proportion
Questions Asked in CAT exam
- The passage given below is followed by four summaries. Choose the option that best captures the essence of the passage.
In the dynamic realm of creativity, artists often find themselves at the crossroads between drawing inspiration from diverse cultures and inadvertently crossing into the territory of cultural appropriation. Inspiration is the lifeblood of creativity, driving artists to create works that resonate across borders. In a globalized era of the modern world, artists draw from a vast array of cultural influences. When approached respectfully, inspiration becomes a bridge, fostering understanding and appreciation of cultural diversity. However, the line between inspiration and cultural appropriation can be thin and easily blurred.
Cultural appropriation occurs when elements from a particular culture are borrowed without proper understanding, respect, or acknowledgment. This leads to the commodification of sacred symbols, the reinforcement of stereotypes, and the erasure of the cultural context from which these elements originated. It is essential to recognize that the impact of cultural appropriation extends beyond the realm of artistic expression, influencing societal perceptions and perpetuating power imbalances.- CAT - 2025
- Para Summary
- The number of distinct integers $n$ for which $\log_{\left(\frac14\right)}(n^2 - 7n + 11)>0$ is:
- CAT - 2025
- Linear Inequalities
- In the set of consecutive odd numbers $\{1, 3, 5, \ldots, 57\}$, there is a number $k$ such that the sum of all the elements less than $k$ is equal to the sum of all the elements greater than $k$. Then, $k$ equals?
- CAT - 2025
- Number Systems
- The number of distinct pairs of integers $(x, y)$ satisfying the inequalities $x>y \ge 3$ and $x + y<14$ is:
- CAT - 2025
- Number Systems
For any natural number $k$, let $a_k = 3^k$. The smallest natural number $m$ for which \[ (a_1)^1 \times (a_2)^2 \times \dots \times (a_{20})^{20} \;<\; a_{21} \times a_{22} \times \dots \times a_{20+m} \] is:
- CAT - 2025
- Linear Inequalities