## Solutions to Fraction decimal questions SSC CGL Set 75

Learn how to solve 10 fraction decimal questions of SSC CGL Set 75 in 15 minutes using special problem solving techniques.

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**SSC CGL level Question Set 75 on fractions decimals indices 7****.**

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### Fraction decimal indices questions solutions SSC CGL Set 75 - time to solve was 15 mins

#### Problem 1.

$\displaystyle\frac{13}{48}$ is equal to,

- $\displaystyle\frac{1}{3+\displaystyle\frac{1}{1+\displaystyle\frac{1}{16}}}$
- $\displaystyle\frac{1}{3+\displaystyle\frac{1}{1+\displaystyle\frac{1}{1+\displaystyle\frac{1}{8}}}}$
- $\displaystyle\frac{1}{2+\displaystyle\frac{1}{1+\displaystyle\frac{1}{8}}}$
- $\displaystyle\frac{1}{3+\displaystyle\frac{1}{1+\displaystyle\frac{1}{2+\displaystyle\frac{1}{4}}}}$

**Solution 1: Problem analysis and solving**

When all four options are continued fractions, evaluating each to the fullest extent will take time. So we **adopt the strategy** of evaluating the choices with smaller depth, that is, first and third options in the first stage because these two can be evaluated faster than the othe two.

#### Technique of evaluation of continued fractions: Example - Option 3

As an example of the process of mentally evaluating a continued fraction let us take up the third choice,

$\displaystyle\frac{1}{2+\displaystyle\frac{1}{1+\displaystyle\frac{1}{8}}}$

**First step:** evaluate the last or lowermost fraction expression,

$1+\displaystyle\frac{1}{8}=\displaystyle\frac{9}{8}$

The *important point at this step* is:

By the nature of continued fractions, till the end result is reached, the result of each evaluation is inverted. This is the key pattern here.

**Second step:** invert result $\displaystyle\frac{9}{8}$ to $\displaystyle\frac{8}{9}$ and the expression to be evaluated now is, $2+\displaystyle\frac{8}{9}$.

The *important point at this step* is the value of 9 in the denominator of the second term. By the addition now, this value won't be changed and at the next step it will go up to the numerator. As our target numerator is 13, we can reject this choice.

**We don't need to evaluate to the end.**

Similarly the first choice ends up with a numerator value of 17 and is rejected.

Likewise, we get numerator value 17 again for the second choice and reject the choice.

Though we know by now that the fourth choice is the answer, still we check it quickly.

$\displaystyle\frac{1}{3+\displaystyle\frac{1}{1+\displaystyle\frac{1}{2+\displaystyle\frac{1}{4}}}}$

=$\displaystyle\frac{1}{3+\displaystyle\frac{1}{1+\displaystyle\frac{1}{\displaystyle\frac{9}{4}}}}$

=$\displaystyle\frac{1}{3+\displaystyle\frac{1}{1+\displaystyle\frac{4}{9}}}$

=$\displaystyle\frac{1}{3+\displaystyle\frac{1}{\displaystyle\frac{13}{9}}}$

=$\displaystyle\frac{1}{3+\displaystyle\frac{9}{13}}$

=$\displaystyle\frac{1}{\displaystyle\frac{48}{13}}$

=$\displaystyle\frac{13}{48}$.

**Answer:** Option d: $\displaystyle\frac{1}{3+\displaystyle\frac{1}{1+\displaystyle\frac{1}{2+\displaystyle\frac{1}{4}}}}$.

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

*--*

**numerator matching***.*

**strategy to speed up solution -- Strategy decision -- solving in mind -- continued fractions**This solution is fast and wholly in mind.

#### Problem 2.

When $\left(\displaystyle\frac{1}{2} -\displaystyle\frac{1}{4}+\displaystyle\frac{1}{5}-\displaystyle\frac{1}{6}\right)$ is divided by $\left(\displaystyle\frac{2}{5} -\displaystyle\frac{5}{9}+\displaystyle\frac{3}{5}-\displaystyle\frac{7}{18}\right)$ the result is,

- $5\displaystyle\frac{1}{10}$
- $3\displaystyle\frac{1}{6}$
- $3\displaystyle\frac{3}{10}$
- $2\displaystyle\frac{1}{18}$

**Solution 2: Problem analysis and strategy decision**

The strategy must be to evaluate both the fraction expressions individually and then divide. Objective is then, how fast we can evaluate the two fraction expressions.

The patterns in both the expressions we discover is the ease of combining the denominators of the positive term pairs and negative term pairs separately. This is the **key pattern** in this problem.

#### Solution 2: Problem solving execution using key pattern

The positive pair of terms in the first and second expressions evaluate respectively to, $\displaystyle\frac{7}{10}$ and $1$.

Similarly the negative pair terms evaluate respectively to,

$-\displaystyle\frac{5}{12}$ and $-\displaystyle\frac{17}{18}$.

The expression values evaluate respectively to,

$\displaystyle\frac{7}{10}-\displaystyle\frac{5}{12}=\displaystyle\frac{17}{60}$, and,

$1-\displaystyle\frac{17}{18}=\displaystyle\frac{1}{18}$.

Dividing, we get the result as,

$\displaystyle\frac{51}{10}=5\displaystyle\frac{1}{10}$.

**Answer.** Option a: $5\displaystyle\frac{1}{10}$.

**Key concepts used:** * Key pattern identification -- Denominator combining ease -- Strategy decision -- Efficient simplification -- Quick solution*.

The solution is reached wholly in mind.

#### Problem 3.

On simplification, $3034-(1002\div{20.04})$ is equal to,

- $2993$
- $2984$
- $3029$
- $2543$

**Solution 3: Problem solving execution**

Taking care of the decimal point in the denominator of the second term, we evaluate the result of the division as,

$\displaystyle\frac{100}{2}=50$.

Subtracting from the first term we get the final result as,

$3034-50=2984$.

**Answer:** Option b. 2984.

**Key conceptys used: ** **Decimal elimination technique -- Quick solution.**

#### Problem** 4.**

Value of $\displaystyle\frac{\displaystyle\frac{5}{3}\times{\displaystyle\frac{7}{51}}\text{ of }\displaystyle\frac{17}{5}-\displaystyle\frac{1}{3}}{\displaystyle\frac{2}{9}\times{\displaystyle\frac{5}{7}}\text{ of }\displaystyle\frac{28}{5}-\displaystyle\frac{2}{3}}$ is,

- $\displaystyle\frac{1}{2}$
- $4$
- $\displaystyle\frac{1}{4}$
- $2$

#### Solution** 4: Problem analysis and solving execution**

Knowing that the $\text{of}$ operator is equivalent to multiplication we evaluate first the product terms in the numerator and denominator. Cancelling out the common factors the products evaluate for the numerator and denominator respectively to,

$\displaystyle\frac{5}{3}\times{\displaystyle\frac{7}{51}}\times{\displaystyle\frac{17}{5}}=\frac{7}{9}$, and,

$\displaystyle\frac{2}{9}\times{\displaystyle\frac{5}{7}}\times{\displaystyle\frac{28}{5}}=\frac{8}{9}$

Carrying out the subtractions,

$\displaystyle\frac{\displaystyle\frac{7}{9}-\displaystyle\frac{1}{3}}{\displaystyle\frac{8}{9}-\displaystyle\frac{2}{3}}$

$=\displaystyle\frac{\displaystyle\frac{4}{9}}{\displaystyle\frac{2}{9}}$

$=2$

**Answer:** Option d: $2$.

**Key concepts used:** ** Fraction arithmetic **--

*.*

**BODMAS rule**#### Problem** 5.**

Value of $\displaystyle\frac{9|3-5|-5|4|\div{10}}{-3(5)-2\times{4}\div{2}}$ is,

- $\displaystyle\frac{9}{10}$
- $\displaystyle\frac{4}{7}$
- $-\displaystyle\frac{8}{17}$
- $-\displaystyle\frac{16}{19}$

#### Solution 5: Problem analysis and solving execution

Taking note of the absolute value operator, the term, $|3-5|$ evaluates to 2 (and not $-2$) so that the numerator, following BODMAS rule, evalutes to, $18-2=16$.

Similarly the denominator evaluates to,

$-15-4=-19$, and the final result thus is,

$-\displaystyle\frac{16}{19}$.

**Answer:** Option d: $-\displaystyle\frac{16}{19}$.

**Key concepts used:** * Fraction arithmetic *--

*--*

**Absolute value operator****.**

*BODMAS rule*#### Problem 6.

The value of

$\left(1+\displaystyle\frac{1}{10+\displaystyle\frac{1}{10}}\right)\left(1+\displaystyle\frac{1}{10+\displaystyle\frac{1}{10}}\right)-$

$\left(1-\displaystyle\frac{1}{10+\displaystyle\frac{1}{10}}\right)\left(1-\displaystyle\frac{1}{10+\displaystyle\frac{1}{10}}\right)$ ÷

$\left[\left(1+\displaystyle\frac{1}{10+\displaystyle\frac{1}{10}}\right)-\left(1-\displaystyle\frac{1}{10+\displaystyle\frac{1}{10}}\right)\right]$ is,

- $2$
- $\displaystyle\frac{90}{101}$
- $\displaystyle\frac{20}{101}$
- $\displaystyle\frac{100}{101}$

#### Solution** 6: Problem analysis, pattern identification and problem solving execution**

We need to consider the two **unique factor expressions as two dummy variables** $a$ and $b$ to transform the given expression to,

$(a^2-b^2)\div{(a-b)}=(a+b)$, where,

$a=\left(1+\displaystyle\frac{1}{10+\displaystyle\frac{1}{10}}\right)$, and

$b=\left(1-\displaystyle\frac{1}{10+\displaystyle\frac{1}{10}}\right)$.

Identification and visualization of the factors is the key pattern identification in this problem.

Adding the two complex factors cancels out the fraction terms leaving 2 as the result.

**Answer:** Option a: $2$.

**Key concepts used:** * Pattern identification* --

*.*

**Algebraic simplification -- Substitution technique**Done easily in mind.

#### Problem** 7.**

The value of $8\displaystyle\frac{1}{2}-\left[3\displaystyle\frac{1}{4}\div{\left\{1\displaystyle\frac{1}{4}-\displaystyle\frac{1}{2}\left(1\displaystyle\frac{1}{2}-\displaystyle\frac{1}{3}-\displaystyle\frac{1}{6}\right)\right\}}\right]$ is

- $4\displaystyle\frac{1}{2}$
- $\displaystyle\frac{2}{9}$
- $4\displaystyle\frac{1}{6}$
- $9\displaystyle\frac{1}{2}$

#### Solution 7: Problem analysis, BODMAS rule and faster mixed fraction expression evaluation

This is an evaluation problem of *mixed fraction expression* guided by BODMAS rule.

#### BODMAS rule

**General evaluation sequence:** from left to right (LTR, just like reading sequence of a sentence in majority of natural languages). In an expression comprising of equal precedence operations only, this rule comes into play.

**B**: **Brackets:** expression enclosed by "[" higher precedence than expression enclosed by "{". Again expression enclosed by "{" gets higher precedence than expression enclosed by "(". As a rule, "[" encloses "{" which encloses "(". Innermost expression enclosed by "(" is evaluated first.

**O**: **Orders** or powers: We also know this as exponentials or indices. Example, $2^2\times{3}$. This has lower precedence than brackets but higher precedence than Division, Multiplication, Addition, Subtraction. Occasionally Overline, a line over a few consecutive terms is used for separating out the terms into an expression, with an effect like a first bracket.

**D M: ** **Division** and **Multiplication**: these have higher precedence than only Addition and Subtraction. Whichever of Division and Multiplication occurs first from left to right sequence will be evaluated first. Occasionally, $\text{ Of }$ operator is used for multiplication.

**A S: ** **Addition** and **Subtraction**: these are of lowest precedence. Whichever of the two occurs first in LTR sequence will be evaluated first.

Conventional approach would be to convert the mixed fractions to the equivalent improper fractions (for example, conversion of $1\displaystyle\frac{1}{2}$ to $\displaystyle\frac{3}{2}$) and proceed with BODMAS rule guided evaluation.

To speed up mixed fraction arithmetic we adopt **Mixed fraction breakup**,

In mixed fraction expression evaluation,we don't convert mixed fraction terms to the equivalent improper fraction form. Instead, we consider each as a two part number: integer plus fraction. Consequently the expression is converted into an integer expression part and a pure proper fraction expression part, speeding up the overall process significantly.

For example, we consider $3\displaystyle\frac{1}{2}-2\displaystyle\frac{1}{3}$ as,

$3+\displaystyle\frac{1}{2}-2 -\displaystyle\frac{1}{3}$

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

Adopting this tactic on an expression involving mixed fractions, we deal with arithmetic operations on simpler proper fractions, thus effectively speeding up evaluation significantly.

#### Solution 7: Problem solving execution

First adding $-\displaystyle\frac{1}{3}$ to $-\displaystyle\frac{1}{6}$ we get $-\displaystyle\frac{1}{2}$. It cancels out the $\displaystyle\frac{1}{2}$ of $1\displaystyle\frac{1}{2}$, leaving $1$ as the outcome.

Next subtracting $\displaystyle\frac{1}{2}$ from $1\displaystyle\frac{1}{4}$ results in $\displaystyle\frac{3}{4}$ which divides $3\displaystyle\frac{1}{4}=\displaystyle\frac{13}{4}$ to result in $\displaystyle\frac{13}{3}$.

Finally, this value of $\displaystyle\frac{13}{3}$ is subtracted from $8\displaystyle\frac{1}{2}=\displaystyle\frac{17}{2}$ to leave $\displaystyle\frac{25}{6}=4\displaystyle\frac{1}{6}$ as the answer.

We have not shown the deduction steps as the need is to evaluate mentally and quickly which is easily met by the process adopted.

Mark that near the end, we have converted mixed fractions to the equivalent improper fraction form to easily carry out first division and then subtraction. This is what we call flexible method. We changed the process slightly according to the need.

**Answer:** Option c: $4\displaystyle\frac{1}{6}$.

**Key concepts used:** ** BODMAS rule **--

*Faster method of mixed fraction expression evaluation -- Mixed fraction breakup technique -- Flexible method.*#### Problem** 8.**

$\sqrt{\displaystyle\frac{4\displaystyle\frac{1}{7}-2\displaystyle\frac{1}{4}}{3\displaystyle\frac{1}{2}+1\displaystyle\frac{1}{7}}\div{\displaystyle\frac{1}{2+\displaystyle\frac{1}{2+\displaystyle\frac{1}{5-\displaystyle\frac{1}{5}}}}}}$ is equal to,

- $1$
- $2$
- $3$
- $4$

#### Solution** 8: Problem analysis**

We feel the problem needs a bit of careful calculation and decided to evaluate the second term of the division operation first and then the numerator and denominator of the first term.

We also note the **key pattern** of LCM of fraction denominators 28 in the numerator and 14 in the denominator of the first term. This should help in simplification. *We will multiply the numerator and denominator by 28 to eliminate fractions of this term at the last stage.*

#### Solution 8: Evaluation of the continued fraction

The last fraction expression evaluates to, $5-\displaystyle\frac{1}{5}=\displaystyle\frac{24}{5}$. By immediate inversion it is transformed to $\displaystyle\frac{5}{24}$ and the expression to be evaluated turns to,

$2+\displaystyle\frac{5}{24}=\displaystyle\frac{53}{24}$, and after inversion again, $\displaystyle\frac{24}{53}$. The next expression to be evaluated turns to,

$2+\displaystyle\frac{24}{53}=\displaystyle\frac{130}{53}$, and after inversion again finally,

$\displaystyle\frac{53}{130}$.

We record this result to lessen the memory load.

Let us show the steps.

$\displaystyle\frac{1}{2+\displaystyle\frac{1}{2+\displaystyle\frac{1}{5-\displaystyle\frac{1}{5}}}}$

$=\displaystyle\frac{1}{2+\displaystyle\frac{1}{2+\displaystyle\frac{1}{\displaystyle\frac{24}{5}}}}$

$=\displaystyle\frac{1}{2+\displaystyle\frac{1}{2+\displaystyle\frac{5}{24}}}$

$=\displaystyle\frac{1}{2+\displaystyle\frac{1}{\displaystyle\frac{53}{24}}}$

$=\displaystyle\frac{1}{2+\displaystyle\frac{24}{53}}$

$=\displaystyle\frac{1}{\displaystyle\frac{130}{53}}$

$=\displaystyle\frac{53}{130}$.

The expression to be evaluated thus simplifies to,

$\sqrt{\displaystyle\frac{4\displaystyle\frac{1}{7}-2\displaystyle\frac{1}{4}}{3\displaystyle\frac{1}{2}+1\displaystyle\frac{1}{7}}\times{\displaystyle\frac{130}{53}}}$.

#### Solution 8: Evaluation of first term numerator and denominator by mixed fraction breakup

The numerator is,

$4\displaystyle\frac{1}{7}-2\displaystyle\frac{1}{4}$

$=3-2+\left(\displaystyle\frac{8}{7}-\displaystyle\frac{1}{4}\right)$

$=1+\displaystyle\frac{25}{28}$.

Similarly the denominator,

$3\displaystyle\frac{1}{2}+1\displaystyle\frac{1}{7}$

$=4+\displaystyle\frac{9}{14}$.

Multiplying both by 28 to eliminate fractions the numerator turns out to 53 and the denominator, 130. This cancels out the second term numerator and denominator leaving 1 as the answer.

With accurate confident evaluations this problem should take just about a minute's time.

**Answer:** Option a: 1.

**Key concepts used:** * Mixed fraction* --

**Key pattern identification --***--*

**Continued fraction***.*

**Mixed fraction breakup technique**#### Problem** 9.**

The value of $\displaystyle\frac{1+\displaystyle\frac{1}{2}}{1-\displaystyle\frac{1}{2}}\div{\displaystyle\frac{4}{7}\left(\displaystyle\frac{2}{5}+\displaystyle\frac{3}{10}\right)}\text { of }\displaystyle\frac{\displaystyle\frac{1}{2}+\displaystyle\frac{1}{3}}{\displaystyle\frac{1}{2}-\displaystyle\frac{1}{3}}$ is,

- $37\displaystyle\frac{1}{2}$
- $18\displaystyle\frac{3}{8}$
- $\displaystyle\frac{3}{2}$
- $\displaystyle\frac{2}{3}$

#### Problem** analysis and solving execution**

There are three main fraction terms separated by first, a division and second, a multiplication operation (the "$\text {of}$" operation is equivalent to multiplication). We will evaluate these three terms sequentially.

The first term,

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

$=\displaystyle\frac{\displaystyle\frac{3}{2}}{\displaystyle\frac{1}{2}}$

$=3$.

The second term,

$\displaystyle\frac{4}{7}\left(\displaystyle\frac{2}{5}+\displaystyle\frac{3}{10}\right)$

$=\displaystyle\frac{4}{7}\times{\displaystyle\frac{35}{50}}$

$=\displaystyle\frac{2}{5}$

And the third term,

$\displaystyle\frac{\displaystyle\frac{1}{2}+\displaystyle\frac{1}{3}}{\displaystyle\frac{1}{2}-\displaystyle\frac{1}{3}}$

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

$=5$.

The target expression thus turns out to,

$3\div{\displaystyle\frac{2}{5}}\times{5}$

$=\displaystyle\frac{75}{2}$

$=37\displaystyle\frac{1}{2}$.

Mark that because of equal precedence of division and multiplication in BODMAS rule, and as expressions are evaluated from left to right, the division operation appears first and inverts $\displaystyle\frac{2}{5}$ to $\displaystyle\frac{5}{2}$ and only then the multiplication by 5 comes into play.

This is a good example of how * left to right sequence of evaluation* determines

*in case of two operations of*

**effective precedence***according to BODMAS rule apppearing one after the other in an expression.*

**equal precedence****Answer.** Option a: $37\displaystyle\frac{1}{2}$.

**Key concepts used:** **BODMAS rule -- Fraction arithmetic -- Left to right sequence of expression evaluation -- Effective precedence.**

We write down the three main term values only.

#### Problem** 10.**

When simplified, the expression $(100)^{\frac{1}{2}}\times{(0.001)^{\frac{1}{3}}}-(0.0016)^{\frac{1}{4}}\times{3^0}+\left(\displaystyle\frac{5}{4}\right)^{-1}$ is equal to,

- $1.0$
- $1.6$
- $0$
- $0.8$

#### Solution 10: Problem analysis and solving

The first term of the first product is equivalent to square root of 100, that is 10. The second term of the product similarly is equivalent to,

$(10^{-3})^{\frac{1}{3}}$

$=10^{-1}$.

Thus after multiplication by 10, the result of the first product turns out to 1.

The first term of the second product is equivalent to,

$\left(2^4\times{10^{-4}}\right)^{\frac{1}{4}}$

$=0.2$.

The value of the second term of the product being $3^0=1$, the second product value turns out to, $0.2$.

The third term is just an inversion and results in,

$\displaystyle\frac{4}{5}$.

So the target expression value is,

$1-0.2+0.8=1.6$.

**Answer:** Option b: $1.6$.

**Key concepts used:** * Indices -- Decimals -- fraction powers -- Decimal elimination technique -- Efficient simplification*.

To simplify evaluation of the fraction powers on decimals, first we express the decimals as a product of integer and pure decimal in negative powers of 10 and then apply the power to the whole.

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