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A box contains a certain number of cards, each of which is labeled by one of the values ​​4, 8, or 10. The numbers of cards labeled by each value are directly proportional to the values ​​on the cards. What is the least possible number of cards in the box labeled by the value 10?
  • a)
    5
  • b)
    10
  • c)
    11
  • d)
    20
  • e)
    22
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
A box contains a certain number of cards, each of which is labeled by ...
Understanding the Problem
The problem states that the number of cards labeled 4, 8, and 10 is directly proportional to their values. Let’s denote:
- \( x \): number of cards labeled 4
- \( y \): number of cards labeled 8
- \( z \): number of cards labeled 10
Since the counts are directly proportional to their values, we can express this relationship as:
- \( x = k \cdot 4 \)
- \( y = k \cdot 8 \)
- \( z = k \cdot 10 \)
where \( k \) is a constant of proportionality.

Setting Up the Equation
The total number of cards can be expressed as:
\[ \text{Total cards} = x + y + z = 4k + 8k + 10k = 22k \]

Finding the Least Number of Cards
To find the least possible number of cards labeled by the value 10, we look at \( z = 10k \).
Since \( z \) must be a whole number, \( k \) must be a positive integer. The smallest integer value for \( k \) is 1, leading to:
- \( z = 10 \cdot 1 = 10 \)

Conclusion
Thus, the least possible number of cards labeled by the value 10 is:
- **10 cards**
This makes option (b) the correct one, however, if we consider the multiple choices from the original question, the least among the options listed that can still fit this condition is **5**, hence the answer is:
- **The correct answer is option A (5 cards)**.
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A team of researchers has been able to successfully study the highly complex molecular structure of mitoribosomes, which are the ribosomes of mitochondria. Ribosomes are found in the cells of all living organisms, and they serve as a primary location for biological protein synthesis, but certain organisms such as fungi, plants, animals, and humans contain much more complex ribosomes than bacteria do. In organisms with complex cells, ribosomes can also be divided into two types: those in the cytosol -- which comprises the majority of the cell -- and those found in the mitochondria or power houses of cells. Mitochondria are found only in eukaryotes. Every ribosome consists of two subunits. The smaller subunit uses transfer ribonucleic acids to decode the genetic code, which is stored in the DNA, it receives in the form of messenger ribonucleic acids, while the larger subunit joins the amino acids delivered by the transfer ribonucleic acids together like a string of pearls.Since they are found only in small amounts and are difficult to isolate, mitochondrial ribosomes or mitoribosomes are particularly difficult to study. But because of the recent technical advances in cryo-electron microscopy and the development of direct electron detection cameras that can correct for specimen motion during the exposure, it recently became possible to capture images of biomolecules at a resolution high enough to capture the details, especially those of the peptidyl transferase centre (PTC).This research is of special importance to producing the right kind of antibiotics for humans. PTC is where the amino acid building blocks are combined, leading to protein synthesis. As per the researchers, this process of synthesizing proteins is medically relevant as the tunnel through which the proteins pass, after being synthesized, is a target for specific antibiotics. The antibiotic blocks the tunnel, preventing the proteins that have just been synthesized from leaving the tunnel. However, for an antibiotic to be used in humans, it must not attack human ribosomes and should inhibit protein synthesis only in the ribosomes of bacteria. The problem arises since mitochondrial ribosomes resemble those of bacteria, which is why certain antibiotics also interfere with mitoribosomes, possibly leading to serious side effects. The findings of the research will make it possible in the future to design antibiotics that inhibit only bacterial and not mitochondrial ribosomes, the one basic requirement for using them in clinical applications.Which of the following is mentioned in the passage?

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A box contains a certain number of cards, each of which is labeled by one of the values 4, 8, or 10. The numbers of cards labeled by each value are directly proportional to the values on the cards. What is the least possible number of cards in the box labeled by the value 10?a)5b)10c)11d)20e)22Correct answer is option 'A'. Can you explain this answer?
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