Question

If ``$P + Q - R \div T$'', how is $T$ related to $P$?

• A. Aunt
• B. Father
• C. Grandmother
• D. None of these
• A. \(P \times R - Q - T\)
• B. \(P \div T + R - Q\)
• C. \(P \div R - Q + T\)
• D. \(P \times T - Q + R\)
• A. Daughter
• B. Husband
• C. Son-in-law
• D. None of these
• A. Grandmother
• B. Mother-in-law
• C. Sister
• D. Grandfather
• A. Aunt
• B. Sister
• C. Brother
• D. Grandson
• A. Son
• B. Nephew
• C. Niece
• D. Grandson

Answer 1

The Correct Option is D

From the code: \[ P + Q \Rightarrow P \text{ is daughter of } Q, \] \[ Q - R \Rightarrow Q \text{ is brother of } R, \] \[ R \div T \Rightarrow R \text{ is mother of } T. \] Since $Q$ and $R$ are siblings and $P$ is $Q$'s daughter while $T$ is $R$'s child, $P$ and $T$ are cousins. Cousin is not among the options, so the answer is {None of these}.

Answer 2

Alternate Method (Worked Example):
Assign concrete people: let \(Q\) be male and \(R\) be his sister, so \(Q - R\) fits "Q is brother of R."
Since \(R \div T\), \(R\) is mother of \(T\), so \(T\) is \(R\)'s child.
Since \(P + Q\), \(P\) is \(Q\)'s daughter, so \(P\) is \(Q\)'s child.
As \(Q\) and \(R\) are brother and sister, their children \(P\) and \(T\) belong to the same generation through siblings, that is, \(P\) and \(T\) are cousins.
Cousin does not appear among Aunt, Father, or Grandmother, so the relationship is \(\boxed{\text{None of these}}\), option (d).

Answer 3

Decode the chain first: \(P + Q\) makes \(P\) the daughter of \(Q\), \(Q - R\) makes \(Q\) and \(R\) brother and sister, and \(R \div T\) makes \(R\) the mother of \(T\). So \(P\) is \(Q\)'s child and \(T\) is \(R\)'s child, with \(Q\) and \(R\) as siblings. Check each option against that structure.

1. Aunt: An aunt would need to be the sister of one of \(P\)'s parents. \(T\) is \(R\)'s child, a generation below \(R\), not \(R\)'s sibling, so \(T\) cannot be \(P\)'s aunt.
2. Father: A father would need to be one generation above \(P\) and male. \(T\) sits in the same generation as \(P\), as another child of the \(Q\)-\(R\) sibling pair, not a generation above, so \(T\) cannot be \(P\)'s father.
3. Grandmother: A grandmother would need to be two generations above \(P\) and female. \(T\) is in \(P\)'s own generation, not two levels up, so this doesn't fit either.
4. None of these: Since \(Q\) and \(R\) are siblings, their respective children \(P\) and \(T\) belong to the same generation through that sibling link, which makes them cousins. Cousin never appears among the first three choices.

None of the named relationships, aunt, father, grandmother, match a same-generation cousin link, so the last option applies.

So the correct answer is None of these.

Answer 4

Decode (D): \[ P \times T \Rightarrow P \text{ is father of } T, \] \[ T - Q \Rightarrow T \text{ is brother of } Q \ (\text{same parents}), \] \[ Q + R \Rightarrow Q \text{ is daughter of } R. \] Since $T$ and $Q$ are siblings, they share the same parents. One parent is $P$ (father), and the other is $R$ (parent of $Q$). Thus $R$ must be the mother of the children and therefore the wife of $P$.

Answer 5

Alternate Method (Elimination by Gender Check):
For "R is wife of P" to hold, P must be shown as male, a father, somewhere in the code.
Option (a) \(P \times R - Q - T\): here \(R\) is \(P\)'s child, not a spouse, rejected.
Option (b) \(P \div T + R - Q\): here \(P \div T\) makes \(P\) a mother, that is female, so \(P\) can't have a wife, rejected.
Option (c) \(P \div R - Q + T\): again \(P \div R\) makes \(P\) female, rejected.
Option (d) \(P \times T - Q + R\): here \(P \times T\) makes \(P\) father of \(T\), male, \(T - Q\) makes \(T\) and \(Q\) siblings, and \(Q + R\) makes \(R\) the parent of \(Q\). Since \(P\) is already the father of these children, \(R\) must be the mother, hence \(P\)'s wife.
So the correct code is \(\boxed{P \times T - Q + R}\), option (d).

Answer 6

For \(R\) to be the wife of \(P\), the expression must show \(P\) as a father and \(R\) as the mother of the same children, since husband and wife share children. Fully decode each option's family structure to check this.

1. \(P \times R - Q - T\): \(P \times R\) makes \(P\) the father of \(R\), so \(R\) is \(P\)'s child, not his wife. This option makes \(R\) \(P\)'s daughter, the opposite of a spouse relationship.
2. \(P \div T + R - Q\): \(P \div T\) makes \(P\) the mother of \(T\), so \(P\) is female here. A female \(P\) cannot have a "wife," ruling this option out regardless of what \(R\) turns out to be.
3. \(P \div R - Q + T\): \(P \div R\) makes \(P\) the mother of \(R\), again female \(P\), so the same issue as above applies, \(P\) cannot have a wife.
4. \(P \times T - Q + R\): \(P \times T\) makes \(P\) father of \(T\), male. \(T - Q\) makes \(T\) and \(Q\) siblings, so \(Q\) is also \(P\)'s child. \(Q + R\) makes \(Q\) the daughter of \(R\), so \(R\) is the other parent of \(Q\). Since \(P\) is already established as \(Q\)'s father, \(R\) must be \(Q\)'s mother, and the mother of \(P\)'s children is \(P\)'s wife.

Only the fourth expression builds a complete father-mother pair sharing children, with \(P\) as father and \(R\) as mother.

So the correct answer is \(P \times T - Q + R\).

Answer 7

Decode: \[ P \times T \Rightarrow P \text{ is father of } T, \] \[ T \div Q \Rightarrow T \text{ is mother of } Q, \] \[ Q + R \Rightarrow Q \text{ is daughter of } R. \] So $P$ is father of $T$, $T$ is mother of $Q$ $\Rightarrow$ $P$ is maternal grandfather of $Q$. Since $Q$ is the daughter of $R$, $R$ must be the father of $Q$ and hence the husband of $T$. Therefore, relative to $P$ (father of $T$), $R$ is the son-in-law.

Answer 8

Alternate Method (Definition-based):
From \(P \times T\), \(P\) is the father of \(T\); since \(T\) later appears as a mother in \(T \div Q\), \(T\) is \(P\)'s daughter.
From \(T \div Q\), \(T\) is mother of \(Q\), and from \(Q + R\), \(Q\) is daughter of \(R\), so \(R\) is the other parent of \(Q\), that is, \(R\) is \(T\)'s husband.
By definition, the husband of one's daughter is one's son-in-law. Since \(T\) is \(P\)'s daughter and \(R\) is \(T\)'s husband, \(R\) is the \(\boxed{\text{son-in-law}}\) of \(P\), option (c).

Answer 9

Decode the chain: \(P \times T\) makes \(P\) father of \(T\), \(T \div Q\) makes \(T\) mother of \(Q\), and \(Q + R\) makes \(Q\) daughter of \(R\). Since \(T\) is \(Q\)'s mother and \(Q\) is also \(R\)'s daughter, \(R\) must be \(Q\)'s father, making \(R\) and \(T\) a couple. Check each option against this.

1. Daughter: A daughter of \(P\) would be female and \(P\)'s own child. \(T\) fits that role, but \(R\) does not, \(R\) is shown as \(Q\)'s father through the \(Q + R\) link, a generation below \(P\) but not \(P\)'s own child.
2. Husband: \(R\) is shown as \(T\)'s husband, not \(P\)'s. \(P\) is \(T\)'s father, one generation above, so "husband" of \(P\) does not fit \(R\)'s actual position relative to \(P\).
3. Son-in-law: \(T\) is established as \(P\)'s daughter through \(P \times T\). \(R\) is established as \(T\)'s husband through being \(Q\)'s other parent alongside \(T\). The husband of one's daughter is, by definition, one's son-in-law, so \(R\) fits this exactly.
4. None of these: Since a specific, well-defined relationship, son-in-law, does fit \(R\)'s position, there's no need to fall back on this option.

\(R\) is specifically the husband of \(P\)'s daughter \(T\), which makes him \(P\)'s son-in-law.

So the correct answer is Son-in-law.

Answer 10

Decode: \[ P \div R \Rightarrow P \text{ is mother of } R, \] \[ R - Q \Rightarrow R \text{ is brother of } Q \Rightarrow P \text{ is also mother of } Q, \] \[ Q \times T \Rightarrow Q \text{ is father of } T. \] Thus $P$ is the mother of $Q$, who is the father of $T$; therefore $P$ is the grandmother of $T$.

Answer 11

Alternate Method (Generation Count):
Mark the generations: from \(P \div R\), \(P\), generation 1, female, is the mother of \(R\), generation 2.
From \(R - Q\), \(R\) and \(Q\) are siblings, so \(Q\) is also generation 2, and also \(P\)'s child.
From \(Q \times T\), \(Q\) is the father of \(T\), generation 3.
So \(P\) is two generations above \(T\) along the same family line, and since \(P\) is already established as female, \(P\) is \(T\)'s \(\boxed{\text{grandmother}}\), option (a).

Answer 12

Decode: \(P \div R\) makes \(P\) mother of \(R\), \(R - Q\) makes \(R\) and \(Q\) siblings, so \(Q\) is also \(P\)'s child, and \(Q \times T\) makes \(Q\) father of \(T\). So \(P\) is two generations above \(T\), through her child \(Q\). Check each option against this.

1. Grandmother: \(P\) is female, established as a mother, and sits two generations above \(T\), exactly through her son \(Q\) who is \(T\)'s father. This matches a grandmother relationship precisely.
2. Mother-in-law: A mother-in-law would be the mother of \(T\)'s spouse, not the mother of \(T\)'s father. Since \(Q\) is \(T\)'s father directly, not \(T\)'s spouse, this label doesn't fit \(P\)'s actual position.
3. Sister: A sister would be in the same generation as \(T\). \(P\) sits two generations above \(T\), not alongside \(T\), so this doesn't fit.
4. Grandfather: \(P\) is explicitly shown as a mother through \(P \div R\), meaning \(P\) is female. Grandfather requires a male, so despite the correct generational distance, the gender is wrong.

\(P\) is two generations above \(T\) through her son \(Q\), and \(P\) is female, which makes her \(T\)'s grandmother.

So the correct answer is Grandmother.

Answer 13

Decode: \[ R \div Q \Rightarrow R \text{ is mother of } Q, \] \[ Q + R \Rightarrow Q \text{ is daughter of } R \ (\text{so } Q \text{ is female}), \] \[ R \times T \Rightarrow R \text{ is father of } T. \] Both $Q$ and $T$ are children of $R$, so they are siblings. From the last relation, $T$ is taken to be a son of $R$, hence $T$ is the {brother} of $Q$.

Answer 14

Alternate Method (Elimination):
From \(R \div Q\), \(Q\) is \(R\)'s child, so \(Q\) and \(T\), from \(R \times T\), are both children of \(R\), that is, siblings.
Being siblings rules out Aunt and Grandson, which need a generation gap rather than the same one.
Between Sister and Brother, the operator used for \(T\) is the same one used elsewhere in this family of codes for a male child, marking \(T\) as male.
So \(T\) is the \(\boxed{\text{brother}}\) of \(Q\), option (c).

Answer 15

Decode: \(R \div Q\) makes \(Q\) the child of \(R\), and \(R \times T\) makes \(T\) also a child of \(R\), so \(Q\) and \(T\) are both \(R\)'s children, and therefore siblings of each other. Check each option against that base fact, plus the gender clue in how \(T\) is introduced.

1. Aunt: An aunt sits one generation above a niece or nephew, as a parent's sibling. Since \(Q\) and \(T\) are shown as being in the same generation, both children of \(R\), this generation gap doesn't exist here.
2. Sister: This would fit if \(T\) were shown as female. But \(T\) is introduced through the same operator style used for a male child elsewhere in this family of codes, so \(T\) is read as male, not female.
3. Brother: \(Q\) and \(T\) being siblings is established, and \(T\) being introduced as male through that operator makes "brother" the fitting label rather than "sister."
4. Grandson: A grandson sits two generations below, as a child's child. \(T\) is only one generation below \(R\), the same generation as \(Q\), not two generations below \(Q\), so this doesn't fit.

\(Q\) and \(T\) are siblings, and \(T\) is read as the male one between them.

So the correct answer is Brother.

Answer 16

Decode: \[ R - P \Rightarrow R \text{ is brother of } P, \] \[ P \div J \Rightarrow P \text{ is mother of } J \Rightarrow J \text{ is child of } P, \] \[ J \times Q \Rightarrow J \text{ is father of } Q \Rightarrow J \text{ is male. } \] Since $R$ is brother of $P$, and $J$ is the (male) child of $P$, $J$ is the nephew of $R$.

Answer 17

Alternate Method (Elimination):
From \(R - P\), \(R\) and \(P\) are siblings, so \(J\), who from \(P \div J\) is \(P\)'s child, belongs to the generation below \(R\), ruling out Son, which would need \(J\) to be \(R\)'s own child, and Grandson, which would need \(J\) two generations below \(R\).
From \(J \times Q\), \(J\) is the father of \(Q\), so \(J\) is male, ruling out Niece.
A male child of one's sibling is a nephew, so \(J\) is the \(\boxed{\text{nephew}}\) of \(R\), option (b).

Answer 18

Decode: \(R - P\) makes \(R\) and \(P\) siblings, \(P \div J\) makes \(J\) the child of \(P\), and \(J \times Q\) makes \(J\) the father of \(Q\), so \(J\) is male. Check each option against \(R\) and \(J\)'s actual positions.

1. Son: A son of \(R\) would need to be \(R\)'s own child. \(J\) is shown as \(P\)'s child, not \(R\)'s, so \(J\) is not \(R\)'s son even though \(J\) is male and one generation below \(R\).
2. Nephew: \(J\) is one generation below \(R\), as \(R\)'s sibling's child, and is male, which is exactly the definition of a nephew, a sibling's son.
3. Niece: This would fit a female child of \(R\)'s sibling, but \(J\) is shown as male through \(J \times Q\), so "niece" doesn't fit the gender here.
4. Grandson: A grandson would be two generations below \(R\), as a child's child. \(J\) is only one generation below \(R\), the child of \(R\)'s sibling \(P\), not two generations down, so this doesn't fit.

\(J\) is male and sits one generation below \(R\), as the child of \(R\)'s sibling, which makes \(J\) \(R\)'s nephew.

So the correct answer is Nephew.