Mathematical Skills Test - 7

Let the number of boys and girls be 7x and 5x, respectively.

Total number of students = 720

7x + 5x = 720

x = 60

Number of girls = 5 × 60 = 300

Number of boys = 7 × 60 = 420

Girls = 300 and boys = 420

So, 120 more girls are required to make the ratio 1 : 1.

Formula:

SI =$\frac{\mathrm{P}\times \mathrm{R}\times \mathrm{T}}{100}$

Where,

P = Principal, R = Rate of interest, T = Time period

Let the sum be Rs. x

Then, according to the simple interest formula,

For 22% per annum,

⇒ SI$=\frac{\mathrm{x}\times 22\times 1}{100}$

And for 20% per annum,

⇒ SI =$\frac{\mathrm{x}\times 20\times 1}{100}$

Now, according to question, difference of 22% per annum and 20% per annum is Rs. 140

$\therefore \frac{\mathrm{x}\times 22\times 1}{100}-\frac{\mathrm{x}\times 20\times 1}{100}=140$

$\Rightarrow \frac{2\times \mathrm{x}}{100}=140$

⇒ 2x = 14000

$\Rightarrow \mathrm{x}=\frac{14000}{2}$

⇒ x = 7000

∴ The sum is Rs. 7000.

Given-

A certain sum of money doubles itself in$15$ years.

Time$T=15$years

Let the principal$P=Rs.X$

Amount$A=Rs.2X$

Simple Interest$SI=A-P$

$\Rightarrow SI=Rs.(2X-X)$

$\Rightarrow SI=Rs.X$

According to the formula-

$R=\frac{100\times SI}{P\times T}$ where$R$ is rate percent per annum

$\Rightarrow R=\frac{100\times X}{X\times 15}$

$\Rightarrow R=\frac{20}{3}$

$\Rightarrow R=6\frac{2}{3}\%$

Given:

The cost price of 48 articles is equal to the selling price of 32 articles

Formula:

Profit = Selling price - Cost price

Profit percent =$\frac{\mathrm{Profit}}{\mathrm{Cost}\mathrm{Price}}$ × 100

According to the question, we have

The cost price of 1 article is$\frac{1}{48}$

The selling price of 1 article is$\frac{1}{32}$

So, Profit = Selling price - Cost Price

⇒ Profit =$\frac{1}{32}-\frac{1}{48}$

LCM of 32 and 48 is 96

⇒ Profit =$\frac{3-2}{96}$

⇒ Profit =$\frac{1}{96}$

Now, The Profit percent is:

Profit percent =$\frac{\mathrm{Profit}}{\mathrm{Cost}\mathrm{Price}}$ × 100

⇒ Profit percent =$\frac{1}{96}$ × 48 × 100

⇒ Profit percent =$\frac{1}{2}$ × 100

⇒ Profit percent = 50%

∴ The profit percent is 50%.

Given:

A, B and C can complete a work in 20 days, 30 days and 60 days respectively.

Total wage paid = Rs. 3000

Formula:

Total work = Efficiency × Time

Calculation:

Efficiency of A in one day =$\frac{1}{20}$

Efficiency of B in one day =$\frac{1}{30}$

Efficiency of C in one day =$\frac{1}{60}$

Ratio of their efficiency =$\frac{1}{20}:\frac{1}{30}:\frac{1}{60}$

Multiply the above ratio by 60, we get

Ratio of their efficiency = 3 : 2 : 1

According to the concept, we have

Ratio of wages = 3 : 2 : 1

Let the wage of A, B and C be 3x, 2x and x respectively

So, 3x + 2x + x = Rs. 3000

⇒ 6x = Rs. 3000

⇒ x = Rs. 500

So, the wage of A = 3x

= Rs. (3 × 500)

= Rs. 1500

And, the wage of B = 2x

= Rs. (2 × 500)

= Rs. 1000

And, the wage of C = x

= Rs. 500

∴ The salary of A, B and C is Rs. 1500, Rs. 1000 and Rs. 500.

Given: one of the angles of a triangle is \(\frac{1}{2}\) radian and the other is 99°.

\(\Rightarrow \angle \mathrm{A}=\left(\frac{1}{2}\right)^{{c}}\) and \(\angle {B}=99^{\circ}\)

Convert the \(\angle {B}\) into radians by multiplying by \(\frac{\pi}{180^{\circ}}\).

\(\Rightarrow \angle B=99^{\circ} \times \frac{\pi}{180^{\circ}}=\frac{11 \pi}{20}\)

As we know, the sum of all angles of a triangle is 180° = π

\(\Rightarrow \angle {A}+\angle {B}+\angle {C}=\pi\)

\(\Rightarrow \angle {C}=\pi-(\angle {A}+\angle {B})\)

\(\Rightarrow \angle {C}=\pi-\left(\frac{1}{2}+\frac{11 \pi}{20}\right)\)

\(\Rightarrow \angle {C}=\pi-\left(\frac{10+11 \pi}{20}\right)\)

\(\Rightarrow \angle {C}=\frac{9 \pi-10}{20}\)

So, one of the angles of a triangle is \(\frac{1}{2}\) radian and the other is 99°, then the third angle in radian measure is\(\angle {C}=\frac{9 \pi-10}{20}\)

Suppose the capacity of the tank = 50 units (LCM of\((\frac{50}{3})\)and 25)

∴ Efficiency of Inlet pipe =\(\frac{50}{(\frac{50}{3})}\)= 3

Efficiency of Outlet pipe= \((\frac{50}{25})\)= 2

Since the cooler is half empty

∴ Quantity of water to be filled = \((\frac{50}{2})\)= 25 units

A person takes the water out for 1 minute and then 2 minute gap that means in 3 minutes, outlet tap is working for only 1 minute.

After 9 minutes∶

Quantity of water filled = 3 × 9 – 2 × 3 = 21 units

10thminute∶ Both inlet and outlet will be working;

∴ Quantity of water filled = 3 – 2 = 1 unit

11thminute∶ Only inlet working;

∴ Quantity of water filled = 3 units

∴ Quantity of water filled up to 11 minutes = 21 + 1 + 3 = 25 units

That means cooler is full in 11 minutes.