Tag: types of ratios

Questions Related to types of ratios

Choose the correct answer form the alternatives given.
The ratio of spirit and water in a mixture is 1: 3. If the volume of the solution is increased by 25% by adding spirit only. What is the resultant ratio of spirit and water? 

maths mixture types of ratios ratios in proportion mathematical logic

  1. $2:3$

  2. $1:4$

  3. $1: 2$

  4. None of these

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Correct Option: A
Explanation:

Let the volume of spirit and water be x and 3x Then, total volume = 4x. Resultant  volume of solution = 1.25 $\times$ 4x = 5x
Therefore, increase in volume = $5x - 4x = x$
So, the new ratio of spirit of water $2x : 3x = 2:3$
It is to be noted that increase in volume is due to addition of spirit only.

The expenses on wheat, meat and vegetables of a family are in the ratio 12: 17 : 3. The prices of these articles are increased by 20%, 30% and 50% respectively. The total expenses of the family on these articles are increased by

maths mixture types of ratios ratios in proportion mathematical logic

  1. 23$\frac{1}{3}$%

  2. 28$\frac{1}{8}$%

  3. 27$\frac{1}{8}$%

  4. 25$\frac{1}{7}$%

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Correct Option: B
Explanation:

Given that expense on Wheat, Meat and Vegetable =12x + 17x + 3x = 32x
New expense on wheat, Meat and Vegetable
= 1.2 $\times 12x + 1.3 \times 17x + 1.5 \times 3x $
= 14.4x + 22.1x + 4.5x = 41x
Percentage increase in expense = $\frac{9}{32} \times 100 = 28\frac{1}{8}$%

The ratio  of number of boys and girls in a school of 720 students is 7 : 5 . How many more girls should be admitted to make the ratio 1 : 1 ?

maths mixture types of ratios ratios in proportion mathematical logic

  1. 100

  2. 120

  3. 80

  4. 150

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Correct Option: B
Explanation:
The ratio of the number of boys to girls is $7:5$.
We make this part to part ratio to part to whole ratio by using the property
$a:b\displaystyle \Rightarrow \dfrac{a}{a+b}:\dfrac{b}{a+b}$
$\displaystyle \therefore $ Ratio of the boys to the total students
=$\displaystyle \dfrac{7}{7+5}=\dfrac{7}{12}$
and the ratio of the girls to the total students
$\displaystyle \dfrac{5}{7+5}=\dfrac{5}{12}$
To get the answer we would first find out the actual number of boys and girls in the school
For this we multiply the total number with their respective ratios
$\displaystyle \therefore $ Number of boys=$\displaystyle \dfrac{7}{12}\times 720=7\times 60=420$
and Number of girls=$\displaystyle \dfrac{5}{12}\times 720=5\times 60=300$
Now we need to obtain the boys to girls ratio as 1:1 For this the number of boys and girls should be equal This can be obtained by adding $420-300=120$ girls in the school

The reciprocal of $\dfrac {-5}{13}$ is _____

composition of ratios types of ratios ratio and proportions ratio and proportion maths

  1. $\dfrac {5}{13}$

  2. $\dfrac {-13}{5}$

  3. $\dfrac {13}{5}$

  4. $\dfrac {-5}{13}$

Show answer
Correct Option: B
Explanation:
The reciprocal (also known as the multiplicative inverse) is the number we have to multiply to get an answer equal to the multiplicative number with recipocal of it is 1.
Then $\frac{-5}{13}\times \frac{-13}{5}=1$.
So recipocal of $\frac{-5}{13}$ is $\frac{-13}{5}$.
So answer is (B) $\frac{-13}{5}$.

 

The subtriplicate ratio of $a : b$ is ____

composition of ratios types of ratios ratio and proportions ratio and proportion maths

  1. $a^{2} : b^{2}$

  2. $a^{3} : b^{3}$

  3. $\sqrt {a} : \sqrt {b}$

  4. $\sqrt [3]{a} : \sqrt [3]{b}$

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Correct Option: D
Explanation:

The subtriplicate ratio of $a : b$ is $\sqrt [3]{a} : \sqrt [3]{b} = (a)^{\frac {1}{3}} : (b)^{\frac {1}{3}}$

If $\dfrac {y}{x-z}=\dfrac{y+x}{z}=\dfrac{x}{y}$ then find $x:y:z$

composition of ratios types of ratios ratio and proportions ratio and proportion maths

  1. $1:2:3$

  2. $3:2:1$

  3. $4:2:3$

  4. $2:4:7$

Show answer
Correct Option: C
Explanation:


$ \dfrac{y}{x-z}=\dfrac{y+x}{z}=\dfrac{x}{y} $

 

Now,

$ \dfrac{y}{x-z}=\dfrac{x}{y} $

$ {{y}^{2}}={{x}^{2}}-xz\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ......(1) $

 

And

$ \dfrac{y+x}{z}=\dfrac{x}{y} $

$ {{y}^{2}}+xy=xz\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ......(2) $

$ {{x}^{2}}-xz+xy=xz $

$ x-z+y=z $

$ 2z=x+y\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ......(3) $

 

$ And $

$ \dfrac{y}{x-z}=\dfrac{y+x}{z} $

$ yz=xy-yz+{{x}^{2}}-xz $

$ 2yz=xy+{{x}^{2}}-xz $

$ 2yz=x\left( y+x \right)-xz $                    [From equation (3)]

$ 2yz=2xz-xz $

$ 2yz=xz $

$ 2y=x $

$ \dfrac{x}{y}=\dfrac{2}{1}\,\,\,\,\,\,\,\,......\,\,\left( 4 \right) $


Substituting this value in equation (3), we get

$ 2z=2y+y $

$ 2z=3y $

$ \dfrac{y}{z}=\dfrac{2}{3}\,\,\,\,\,......\,\,\left( 5 \right) $


By equation (4) and (5), we get

$ x:y:z=4:2:3 .$


Hence, this is the answer.

If $\left( {p - q} \right)\,:\left( {q - x} \right)\,$ be the duplicate ratio of $p:q$, then : $\dfrac{1}{p} + \dfrac{1}{q} = \dfrac{1}{x}$

composition of ratios types of ratios ratio and proportions ratio and proportion maths

  1. True

  2. False

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Correct Option: A
Explanation:
$\left(p-x\right):\left(q-x\right)$ is the duplicate ratio of $p:q$

we know,
 if $a^2 : b^2$ is the duplicate  ratio of $a : b$
         now a/c to question,
$(p -x) : (q - x)$ is the duplicate ratio of $p : q$ 
so, from above rule,
$(p -x ) : (q - x ) = p^2 : q^2$


So,$\dfrac{{p}^{2}}{{q}^{2}}=\dfrac{p-x}{q-x}$

$\Rightarrow\,\dfrac{q-x}{{q}^{2}}=\dfrac{p-x}{{p}^{2}}$

$\Rightarrow\,\dfrac{q}{{q}^{2}}-\dfrac{x}{{q}^{2}}=\dfrac{p}{{p}^{2}}-\dfrac{x}{{p}^{2}}$

$\Rightarrow\,\dfrac{1}{q}-\dfrac{x}{{q}^{2}}=\dfrac{1}{p}-\dfrac{x}{{p}^{2}}$

$\Rightarrow\,\dfrac{1}{q}-\dfrac{1}{p}=\dfrac{x}{{q}^{2}}-\dfrac{x}{{p}^{2}}$

$\Rightarrow\,\dfrac{p-q}{pq}=\dfrac{x\left({p}^{2}-{q}^{2}\right)}{{p}^{2}{q}^{2}}$

$\Rightarrow\,p-q=\dfrac{x\left(p-q\right)\left(p+q\right)}{pq}$

$\Rightarrow\,1=\dfrac{x\left(p+q\right)}{pq}$

$\Rightarrow\,\dfrac{1}{x}=\dfrac{\left(p+q\right)}{pq}$

$\Rightarrow\,\dfrac{1}{x}=\dfrac{1}{q}+\dfrac{1}{p}$

$\therefore\,\dfrac{1}{p}+\dfrac{1}{q}=\dfrac{1}{x}$

Hence the given statement is true.

If $2x=3y$ and $4y=5z$, then $x:z=$

composition of ratios types of ratios ratio and proportions ratio and proportion maths

  1. $4:3$

  2. $8:15$

  3. $3:4$

  4. $15:8$

Show answer
Correct Option: D
Explanation:

Given,

$2x=3y$

or, $\dfrac{x}{y}=\dfrac{3}{2}$.....(1).

Again 

$4y=5z$

or, $\dfrac{y}{z}=\dfrac{5}{4}$.....(2).

Now multiplying (4) and (5) we get,

$\dfrac xy \times \dfrac yz=\dfrac 32 \times \dfrac 54$

$\dfrac{x}{z}=\dfrac{15}{8}$

or, $x:z=15:8$

If $\cfrac{a}{2}=\cfrac{b}{3}=\cfrac{c}{4}$, then $a:b:c=$

composition of ratios types of ratios ratio and proportions ratio and proportion maths

  1. $2:3:4$

  2. $4:3:2$

  3. $3:2:4$

  4. None of these

Show answer
Correct Option: A
Explanation:

Given, $\displaystyle \frac{a}{2} = \frac{b}{3} = \frac{c}{4}$


Lets take $\displaystyle \frac{a}{2} = \frac{b}{3} = \frac{c}{4} = k$


So, $\dfrac{a}{2}  = k$

$a = 2k$

$\dfrac{b}{3}  = k$

$b = 3k$

$\dfrac{c}{4} = k$

$c = 4k$

i.e., $a : b: c = 2k : 3k : 4k$

$a : b; c = 2 : 3 : 4$  

If $a:b=3:4$, then $4a:3b=$

composition of ratios types of ratios ratio and proportions ratio and proportion maths

  1. $4:3$

  2. $3:4$

  3. $1:1$

  4. None of these

Show answer
Correct Option: C
Explanation:

Given $a:b=3:4$

or, $\dfrac{a}{b}=\dfrac{3}{4}$
or, $4a={3b}$.....(1).
Now,
$\dfrac{4a}{3b}$
$=\dfrac{3b}{3b}$ [ Using (1)]
$=\dfrac{1}{1}$.
So $4a:3b=1:1$.