## Solutions to hard algebra questions for SSC CGL: Algebraic problem solving techniques

Learn to solve hard algebra questions for SSC CGL 74 in 12 minutes.

Know how to use algebra problem solving techniques for quick solution.

Take the timed test first like a mock test. Then clarify doubts and consolidate concepts by going through the solutions.

If you have not yet taken the corresponding test yet, take the test at, * SSC CGL level question set 74 on Algebra 16* before going through the solution.

### Quick solutions to 10 hard algebra questions for SSC CGL exam - answering time was 12 mins

**Q1. **If $a:b=2:3$ and $b:c=4:5$, find $a^2:b^2:bc$.

- $16:36:20$
- $4:9:45$
- $4:36:40$
- $16:36:45$

** Solution 1 - Problem analysis and execution**

Given,

$a:b=2:3$,

Or, $a^2:b^2=4:9$, squaring both sides of the equation.

Also given,

$b:c=4:5$,

Or, $b^2:bc=4:5$, multiplying numerator and denominator of LHS by $b$ to equalize the middle term to $b^2$ between the two ratios as a prerequisite for joining them to a single one.

We have transformed the LHSs of the two ratios to have the same common middle term $b^2$ in the denominator of the first ratio and in the numerator of the second ratio.

To join the two ratios now we need to equalize the middle term values corresponding to $b^2$ of the RHSs to their LCM,

$9\times{4}=36$.

Transforming the first ratio RHS,

$a^2:b^2=4:9=16:36$.

Similarly transforming the second ratio RHS,

$b^2:bc=4:5=36:45$.

Now we can join the two ratios,

$a^2:b^2:bc=16:36:45$.

**Answer:** Option d: $16:36:45$.

**Key concepts used:** * Ratio joining* --

*Equalizing middle term of LHSs**.*

**-- Equalizing corresponding middle term values of RHSs****Q2.** If $a+b+c=4\sqrt{3}$ and $a^2+b^2+c^2=16$, then the ratio of $a:b:c$ is,

- $1:\sqrt{2}:\sqrt{3}$
- $1:2:3$
- $1:1:1$
- None of these

**Solution 2 - Problem analysis and execution**

To eliminate the surd term and also to create $(a^2+b^2+c^2)$ from the first given equation, we must square it up to get,

$a+b+c=4\sqrt{3}$,

Or, $(a+b+c)^2=48$.

Bringing the value of the second given equation also to 48 to be able to equate the LHS with $(a+b+c)^2$ we have,

$a^2+b^2+c^2=16$

Or, $3(a^2+b^2+c^2)=48$.

Equating the two,

$(a+b+c)^2=3(a^2+b^2+c^2)$

Or, $2(a^2+b^2+c^2)-2ab -2bc-2ca=0$,

Or, $(a-b)^2+(b-c)^2+(c-a)^2=0$.

As the sum of squares to be zero, each square expression must individually be zero,

$(a-b)^2=(b-c)^2=(c-a)^2=0$.

Or, $(a-b)=(b-c)=(c-a)=0$

Or, $a=b=c$.

So required ratio is,

$a:b:c=1:1:1$.

**Answer:** Option c : $1:1:1$.

**Key concepts used:** **Key pattern identification -- Principle of zero sum of square terms -- Input transformation -- Ratio and proportion****.**

**Note:** In fact, the key pattern has been the potential fruitful relation between the two expressions on the LHSs of the two given equations. Unless we could eliminate the intermediary numeric values and bring the two expressions together we couldn't achieve our objective of using the potential in the relation between them.

**Q3. **If $x^2+y^2+z^2=2(x+z-1)$, then the value of $x^3+y^3+z^3$ is,

- $0$
- $1$
- $2$
- $-1$

**Solution 3 - Problem analysis and execution**

Key pattern identified is the possibility of transforming the given expression to the form of a sum of squares equal to zero from which we will be able to find the values of $x$, $y$ and $z$ directly.

Rearranging the terms of the given expression,

$x^2+y^2+z^2=2(x+z-1)$

Or, $(x-1)^2+y^2+(z-1)^2=0$.

As each term of sum of squares must be zero for the sum to be zero,

$x-1=0$

$y=0$, and

$z-1=0$.

So,

$x=1$, $y=0$, and $z=1$, and

$x^3+y^3+z^3=2$.

**Answer:** Option c: 2.

**Key concepts used: Key pattern identification -- **

**Principle of zero sum of square terms.**

**Q4. **If $a+b+c=0$, then the value of $2b^2c^2+2c^2a^2+2a^2b^2 - a^4-b^4-c^4$ is,

- 7
- 0
- 28
- 14

**Solution 4 - Problem analysis and strategy decision**

We need to use two-step transformation of the given expression, keeping the number of terms of the expresssion to the minimum. Two-step means, we need to raise to the power 2 twice, because of presence of $a^4$ in the target expression. Keeping number of terms minimum means, while squaring we would square a two term expression, as far as possible, not a three term expression.

With this strategy we transform the LHS of the given expression to a two-term expression and then raise the equation to its square,

$a+b+c=0$,

Or, $a+b=-c$,

Or, $a^2+2ab+b^2=c^2$

Or, $a^2+b^2-c^2=-2ab$.

Again raising to the power 2,

$a^4+b^4+c^4+2a^2b^2-2c^2a^2-2b^2c^2=4a^2b^2$

Or, $2b^2c^2+2c^2a^2+2a^2b^2-a^4-b^4-c^4=0$,

**Answer:** Option b: 0.

**Key concepts used: Key pattern identification -- Analytical approach example -- Strategic decision making -- Target driven input transformation**

**.**

**Note:** At the last stage, we raised a three term expression to the power of 2 because by then the solution was clearly visible.

**Q5. **If $\displaystyle\frac{a}{b}=\frac{2}{3}$ and $\displaystyle\frac{b}{c}=\frac{4}{5}$ then the value of the ratio $\displaystyle\frac{a+b}{b+c}$ is,

- $\displaystyle\frac{20}{27}$
- $\displaystyle\frac{8}{6}$
- $\displaystyle\frac{6}{8}$
- $\displaystyle\frac{27}{20}$

**Solution 5 - Problem analysis and execution**

Recognizing a hint of componendo dividendo pattern in the target expression we decide to manipulate the given expressions to make these ready for applying componendo dividendo.

Just adding 1 to both sides of the first given expression,

$\displaystyle\frac{a}{b}=\frac{2}{3}$

Or, $\displaystyle\frac{a+b}{b}=\frac{5}{3}$.

We need to invert the second given expression to bring $b$ in the denominator,

$\displaystyle\frac{b}{c}=\frac{4}{5}$,

Or, $\displaystyle\frac{c}{b}=\frac{5}{4}$,

Now adding 1 to both sides,

$\displaystyle\frac{b+c}{b}=\frac{9}{4}$.

Taking the ratio,

$\displaystyle\frac{a+b}{b+c}=\frac{20}{27}$.

**Answer:** Option a: $\displaystyle\frac{20}{27}$.

**Key concepts used:** **Key pattern identification -- End state analysis -- input transformation -- partially hidden componendo dividendo -- adapted componendo dividendo.**

Not only the possibility of componendo dividendo was partially hidden, we needed to adapt the standard three step componendo dividendo method also.

**Q6.** If $U_n=\displaystyle\frac{1}{n}-\displaystyle\frac{1}{n+1}$, then the value of $U_1+U_2+U_3+U_4+U_5$ is,

- $\displaystyle\frac{1}{2}$
- $\displaystyle\frac{1}{3}$
- $\displaystyle\frac{5}{6}$
- $\displaystyle\frac{2}{5}$

**Solution 6 - Problem analysis and execution**

This is a sum of a series of members each defined by $U_n$.

The key pattern immediately identified is—the second negative term of $U_n$ will be cancelled out by the equal and positive first term of the next member in the series, $U_{n+1}$, leaving just the first term of $U_n$ and the second term of $U_{n+1}$ in their sum,

$U_n+U_{n+1}=\left(\displaystyle\frac{1}{n}-\displaystyle\frac{1}{n+1}\right)+\left(\displaystyle\frac{1}{n+1}-\displaystyle\frac{1}{n+2}\right)$

$=\displaystyle\frac{1}{n}-\displaystyle\frac{1}{n+2}$.

In the target sum of five members then only the first term, $\displaystyle\frac{1}{1}$ and the last term $-\displaystyle\frac{1}{6}$ will be left out. All the 8 other intermediate terms will be cancelled out.

The final result will be,

$E=1-\displaystyle\frac{1}{6}=\displaystyle\frac{5}{6}$.

By enumeration also you can verify the result.

The terms of the target expression are,

$U_1=1-\displaystyle\frac{1}{2}$,

$U_2=\displaystyle\frac{1}{2}-\displaystyle\frac{1}{3}$,

$U_3=\displaystyle\frac{1}{3}-\displaystyle\frac{1}{4}$,

$U_4=\displaystyle\frac{1}{4}-\displaystyle\frac{1}{5}$, and

$U_5=\displaystyle\frac{1}{5}-\displaystyle\frac{1}{6}$.

And their sum will be,

$1-\displaystyle\frac{1}{6}=\displaystyle\frac{5}{6}$.

**Answer:** Option c : $\displaystyle\frac{5}{6}$.

**Key concepts used: **

*.*

**Key pattern identification of cancellation of adjacent terms of consecutive members of the series -- Verification by enumeration -- Instant solution**After you discover the key pattern, solution is instant.

** Q7.** If $a*b=2a+3b-ab$, then the value of $(3*5+5*3)$ is,

- 2
- 4
- 6
- 10

**Solution 7 - Problem analysis and solving execution**

If the coefficients of the variables, $a$ and $b$ were same in the expression for the $*$ operation, $(3*5)$ would have been equal to the special $*$ operation on the two in reverse order, $(5*3)$. In this case, the order of appearance of the two variables in each operation pair is important, and so we have to evaluate each operation pair carefully.

By substituting the values of $a=3$, $b=5$ in the RHS of the given relation, we get the value of $(3*5)$ as,

$(3*5)=6+15-15=6$.

Similarly for evaluating $(5*3)$, $a=5$ and $b=3$, so that,

$(5*3)=10+9-15=4$.

Sum of the two is then, 10.

**Answer:** Option d: 10.

** Key concepts used:** * Key pattern identificatoion* --

**Variable value substitution.**

** Q8.** If $\displaystyle\frac{x}{2x^2+5x+2}=\frac{1}{6}$, then the value of $x+\displaystyle\frac{1}{x}$ is,

- $2$
- $-2$
- $-\displaystyle\frac{1}{2}$
- $\displaystyle\frac{1}{2}$

** Solution 8 - Problem analysis and execution**

Examining the given and target expression, the best way to proceed seemed to cross-multiply the given expression and simplify,

$\displaystyle\frac{x}{2x^2+5x+2}=\frac{1}{6}$,

Or, $2x^2+5x+2=6x$

Or, $2x^2+2=x$

Or, $x+\displaystyle\frac{1}{x}=\frac{1}{2}$, dividing the equation by $2x$.

**Answer:** Option d: $\displaystyle\frac{1}{2}$.

**Key concepts used:** * Problem analysis* --

**Straightforward approach****.****Q9.** If $x=\displaystyle\frac{\sqrt{3}+1}{\sqrt{3}-1}$, and $y=\displaystyle\frac{\sqrt{3}-1}{\sqrt{3}+1}$, then the value of $x^2+y^2$ is,

- $13$
- $0$
- $14$
- $15$

**Solution 9 - Problem analysis, key pattern identification and problem solving**

The useful results that are identified from the given expressions immediately are,

$x=\displaystyle\frac{1}{y}$,

Or, $xy=1$, and

$x+y=\displaystyle\frac{1}{2}\left[(\sqrt{3}+1)^2+(\sqrt{3}-1)^2\right]=4$.

The denominator in $(x+y)=\displaystyle\frac{\sqrt{3}+1}{\sqrt{3}-1}+\displaystyle\frac{\sqrt{3}-1}{\sqrt{3}+1}$ is,

$(\sqrt{3}-1)(\sqrt{3}+1)=2$.

And in the numerator of sum of squares the surd middle terms cancel out resulting in,

$(\sqrt{3}+1)^2+(\sqrt{3}-1)^2=2\times{(3+1)}=8$.

Dividing the two results we get,

$x+y=\displaystyle\frac{8}{2}=4$.

So instead of evaluating squares of $x$ and $y$ and summing up, we use the above two values in the target expression,

$x^2+y^2=(x+y)^2-2xy=16-2=14$.

**Answer:** Option c: $14$.

**Key concepts used:** **Key pattern identification -- Surd algebra -- ****Efficient simplification.**

** Q10.** If $\displaystyle\frac{2x-y}{x+2y}=\displaystyle\frac{1}{2}$, then the value of $\displaystyle\frac{3x-y}{3x+y}$ is,

- $1$
- $\displaystyle\frac{3}{5}$
- $\displaystyle\frac{1}{5}$
- $\displaystyle\frac{4}{5}$

**Solution 10 - Problem analysis, key pattern identification and problem solving**

The signature pattern of componendo dividendo is identified in the target expression, but no straightforward way could be seen to transform the given expression in the form of its pair, $\displaystyle\frac{3x}{y}$.

We took next best approach to cross-multiply the two sides of the given expression,

$\displaystyle\frac{2x-y}{x+2y}=\displaystyle\frac{1}{2}$,

Or, $4x-2y=x+2y$.

Or, $3x=4y$,

Or, $\displaystyle\frac{3x}{y}=4$, this is the fraction pattern pair we were looking for.

Now we can carry out the three-step componendo dividendo to get the final result directly,

$\displaystyle\frac{3x-y}{3x+y}=\frac{4-1}{4+1}=\frac{3}{5}$.

Our target was all along to transform the given expression to the form of $\displaystyle\frac{3x}{y}$ in the LHS, so that we can carry out the componendo dividendo on it.

**Answer: **Option b: $\displaystyle\frac{3}{5}$.

**Key concepts used:** * Partially hidden componendo dividendo* --

*--*

**Target driven input transformation**

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