26 practice questions on simplifying-algebraic-expressions from the Expressions using Letter-Numbers section of the UPSC Prelims syllabus. 26 come with a written explanation. Try the sample set below - the answer stays hidden until you ask for it.
Simplify $10y-4y$.
Correct answer: A - $6y$
Taking four lots of $y$ away from ten lots leaves six lots, which is $6y$. Adding instead gives $14y$, dropping the letter gives $6$, and $40y$ multiplies the coefficients.
Simplify $10-(x+4)$.
Correct answer: D - $6-x$
The minus sign in front of the bracket applies to both terms inside, so $10-(x+4)=10-x-4=6-x$. Ignoring the sign on the $4$ gives $14-x$ or $10-x+4$, and $6+x$ keeps the wrong sign on $x$.
Read these two statements. Statement I: $6m$ and $m$ are like terms. Statement II: $6m$ and $6n$ are like terms. Which of them is correct?
Correct answer: C - Only I
Both $6m$ and $m$ carry the letter $m$, so they are like terms and Statement I is true. The terms $6m$ and $6n$ carry different letters, so they stay unlike however similar their coefficients look, making Statement II false.
Check the simplification $4a+3a=7a$ by taking $a=2$. What values do the two sides give?
Correct answer: A - $14$ and $14$
With $a=2$ the left side is $4\times 2+3\times 2=8+6=14$, and the right side is $7\times 2=14$, so the two agree and the simplification is correct. None of the mismatched pairs can come from this substitution.
How can you check whether $3x+2x$ really simplifies to $5x$?
Correct answer: C - Put the same number in place of $x$ in both expressions and compare the values
Two expressions are the same only when they give equal values for every value of the letter, so substituting a number is a genuine test. Counting letters or terms says nothing about value, and adding a coefficient to a letter is not a valid operation.
Which expression is already fully simplified, with no like terms left to combine?
Correct answer: B - $5x+3y-2$
In $5x+3y-2$ the three terms carry $x$, $y$ and no letter, so no two of them can be joined. The second has $5x$ and $3x$, the third has $4y$ and $-y$, and the fourth has the constants $8$ and $3$, each of which can still be combined.
Terms that carry different letters are called:
Correct answer: D - Unlike terms
Terms count as alike only when their letter parts match, so terms with different letters are unlike terms. Constant terms carry no letter at all, and a coefficient is simply the number written in front of a letter.
Rani simplifies $2(x+5)$ to $2x+5$. Substituting $x=1$ into both, what does she find?
Correct answer: B - $12$ and $7$, so her simplification is wrong
With $x=1$ the original gives $2\times 6=12$ while her answer gives $2+5=7$, and the two disagree, so she is wrong; the correct expansion is $2x+10$. The matching pairs cannot arise here, and the original never gives $6$.
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