Tag: maths

Questions Related to maths

The difference of the squares is of two numbers is 80% of the sum of their squares The ratio of the larger number to the smaller number is

maths mixture types of ratios ratios in proportion mathematical logic

  1. 5 : 2

  2. 2 : 5

  3. 3 : 1

  4. 1 : 3

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Correct Option: C
Explanation:

Let the two numbers be x and y Then $\displaystyle x^{2}-y^{2}=80\%$ of $\displaystyle (x^{2}+y^{2})$
$\displaystyle \Rightarrow x^{2}-y^{2}=\frac{4}{5}(x^{2}+y^{2})\Rightarrow x^{2}-\frac{4}{5}x^{2}=\frac{4}{5}y^{2}+y^{2}$
$\displaystyle \Rightarrow \frac{1}{5}x^{2}=\frac{9}{5}y^{2}\Rightarrow \frac{x^{2}}{y^{2}}=\frac{9}{1}\Rightarrow \frac{x}{y}=\frac{3}{1}\Rightarrow x:y=3:1$

If a, b, c, d are positive real number such that $\frac {a}{3}=\frac {a+b}{4}=\frac {a+b+c}{5}=\frac {a+b+c+d}{6}$, then $\frac {a}{b+2c+3d}$ is

maths mixture types of ratios ratios in proportion mathematical logic

  1. $\frac {1}{2}$

  2. 1

  3. 2

  4. not determinable

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Correct Option: A
Explanation:

$a =3k$
$b=k$
$c= 5k-4k =k$
$d =6k-5k =k$
$\frac {a}{b+2c+3d}=\frac {3k}{k+2k+3k}=\frac {1}{2}$

The incomes of A, B and C are in the ratio 7 : 9 : 12 and their spendings are in the ratio 8 : 9 : 15. If A saves $\displaystyle \left ( 1/4 \right )^{th}$ of his income then?

maths mixture types of ratios ratios in proportion mathematical logic

  1. $56 : 99 : 69$

  2. $69 : 56 : 99$

  3. $99 : 56 : 69$

  4. $99 : 69 : 56$

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Correct Option: A
Explanation:

Solution:

Let income of $A=7x$
Income of $B=9x$
Income of $C=12x$
and  Spendings of $A=8y$
Spendings of $B=9y$
Spendings of $C=15y$
Income=Savings + Expenditures
Savings of $A=\cfrac14\times 7x=\cfrac{7x}4$
or, $7x=\cfrac{7x}{4}+8y$
or, $21x=32y$
or, $x=\cfrac{32}{21}y$
Now,
Income of $A=7x=7\times \cfrac{32}{21}y=\cfrac{32}{3}y$
Income of $B=9x=9\times \cfrac{32}{21}y=\cfrac{96}{7}y$
Income of $C=7x=12\times \cfrac{32}{21}y=\cfrac{128}{7}y$
Now, 
Savings of $A=7x-8y=\cfrac{32}3y-8y=\cfrac83y$
Savings of $B=9x-9y=\cfrac{96}7y-9y=\cfrac{33}7y$
Savings of $C=12x-15y=\cfrac{128}7y-15y=\cfrac{23}7y$
So, Savings of $A,B$ and $C$ in ratio
$\cfrac83:\cfrac{33}{7}:\cfrac{23}{7}::56:99:69$
Hence, A is the correct option.

The least whole number which when subtracted from both the terms of the ratio  $6:7$  gives a ratio less than $16:21.$

maths mixture types of ratios ratios in proportion mathematical logic

  1. $2$

  2. $3$

  3. $4$

  4. $6$

Show answer
Correct Option: B
Explanation:

Let the whole number is X.
Now, according to question,
(6-X) / (7-X) < 16/21
21 *(6-X) < 16 *(7-X)
126 - 21X < 112 - 16X
126 - 112 < -16X + 21X
14 < 5X
5X > 14
X > 2.8
So, Least such whole number would be 3.

The length of the ribbon was originally $30cm$. It was reduced in the ratio $5:3$. What is its length now?

maths mixture types of ratios ratios in proportion mathematical logic

  1. $15$

  2. $18$

  3. $20$

  4. $25$

Show answer
Correct Option: B
Explanation:

Length of ribbon originally $=30cm$
Let the original length be $5x$ and reduced length be $3x$.
But $5x=30cm$
$\Longrightarrow x=\dfrac{30}{5}cm=6cm$
Therefore, reduced length $=3\times6cm=18cm$

State whether true or false:
The following operation will increase the value of the original fraction:
Multiply a positive proper fraction by $\cfrac{3}{8}$.

maths mixture types of ratios ratios in proportion mathematical logic

  1. True

  2. False

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Correct Option: B
Explanation:

Decrease: Multiplying a  proper fraction by a value less than 1 (0 < x < 1) decreases the number.

So here $3/8 = 0.375 < 1$ So, the value of original fraction decreases.

State whether true or false:
Multiplying the numerator of a positive proper fraction by $\cfrac{3}{2}$ will cause the original value to increase.
maths mixture types of ratios ratios in proportion mathematical logic

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

Multiplying any fraction by a value greater than 1 will increase its value.

Here, $3/2 = 1.5 > 1$. So, the given statement is True.

State whether true or false:
The following operation will increase the value of the original fraction.
Divide a positive, proper fraction by $\cfrac{3}{13}$

maths mixture types of ratios ratios in proportion mathematical logic

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

Increase: Dividing a positive number by a positive, proper fraction increases the number.

State whether true or false
The given operation will increase the value of the original fraction.
Adding 1 to the numerator of a positive proper fraction and subtracting 1 from its denominator.

maths mixture types of ratios ratios in proportion mathematical logic

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

Increase: As the numerator of a positive, proper fraction increases, the value of the fraction increases. As the denominator of a positive, proper fraction decreases, the value of the fraction also increases. Both actions will work to increase the value of the fraction.

State whether true or false:
The following operation will increase the value of the original fraction.
Multiply both the numerator and denominator of a positive proper fraction by $3\cfrac{1}{2}$.

maths mixture types of ratios ratios in proportion mathematical logic

  1. True

  2. False

Show answer
Correct Option: B
Explanation:

Stay the same: Multiplying or dividing the numerator and denominator of a fraction by the same number will not change the value of the fraction.