Test: Linear Equations in One Variable - 1 - ACT MCQ

Let the number be x.

According to Question,

2x/3 - 50 = x/4 + 40

⇒ 5x/12 = 90

⇒ x = (90 x 12)/5

⇒ x = 216

∴ The number is 216.

Given:

Number of Five rupee note = Number of Ten rupee note = Number of Twenty rupee note

Calculation:

Let the equal number of five, ten and twenty rupee notes be x

⇒ 5x + 10x + 20x = 385

⇒ 35x = 385

⇒ x = 385/35 = 11

Total number of notes = 11 notes of Rs 5 + 11 notes of Rs 10 + 11 Rs of 20 =  33 notes

Check:

11 notes of Rs 5 + 11 notes of Rs 10 + 11 Rs of 20 = 11 × 5 + 11 × 10 + 11 × 20 = 55 + 110 + 220 = Rs. 385

Total number of notes = 11 × 3 = 33 

Therefore the correct answer is 33.
Alternate Method:

Since the number of Rs 5, Rs 10, and Rs 20 notes are equal, the ratio of the number of notes is = 1 ∶ 1 ∶ 1

Or, we can say, the total number of notes must be multiple of 3.

 Only two options are multiple of 3, others can be eliminated.

option 2 : 33. That means the number of notes of each dimension is 33/3 = 11.

5 × 11 + 11 × 10 + 11 × 20 = 385

∴ The total number of notes is 33.

Let the numbers be ‘x’ and ‘y’

Given, sum = x + y = 34

Product = xy = 288

we need to find   x³ + y³

As we know, x³ + y³ = (x + y)³ – 3xy(x + y) = (x + y)[(x + y)² – 3xy]

put the values of x + y and xy in to the equation

∴ Sum of their cubes = 34 × [(34)² – 3 × 288] = 34 × (1156 – 864) = 34 × 292 = 9928

Given:

x⁶ + x⁵ + x⁴ + x³ + x² + x + 1 = 0

Calculation:

Considering the given equation

x⁶ + x⁵ + x⁴ + x³ + x² + x + 1 = 0      -----(1)

In equation (1) multiplying by x

x⁷ + x⁶ + x⁵ + x⁴ + x³ + x² + x = 0      -----(2)

Equation (2) – (1)

⇒ x⁷ - 1 = 0

⇒ x⁷ = 1

x⁵⁰⁵⁴ + x⁶⁰⁵⁵ - 7

⇒ (x⁷)⁷²² + (x⁷)⁸⁶⁵ – 7

⇒ (1)⁷²² + (1)⁸⁶⁵ – 7

⇒ 1 + 1 – 7

⇒ - 5

∴ Required value is – 5

⇒ Milk used to make 200 gms of panner = 1 litre

⇒ Milk used to make 20 kgs paneer = (20 × 1000/200) × 1 = 100 litre

⇒ Cost of 1 litre milk = Rs. 40

⇒ Cost of 100 litre Milk = 100 × 40 = Rs. 4000

1 student reads = 5 subjects

1300 students read = 5 × 1300 = 6500 subjects

1 subject is read by = 65 students

So, total different subjects read = 6500/65 = 100

Calculation:

Let the unit digit be x and tens digit be y

According to the question,

⇒ 10y + x - (10x + y) = 18

⇒ 9y - 9x = 18

⇒ y - x = 2

It means that difference of digits of two-digit numbers is 2.

∴ Six cases other than (13, 31) are possible

(24, 42) (35, 53) (46, 64) (57, 75) (68, 86) (79, 97)
Mistake Points:
Here we cannot take (20, 2)
Because 2 is not a two-digit number. Normally we do not write numbers from 1 - 9  as 01, 02, 03, 04, 05, 06, 07, 08, and 09.

Given:

The given equation is 10x + 26 = 0

Concept:

Rational numbers are the numbers that can be written in the form of p/q, where q is not equal to zero. 

Calculation:

10x + 26 = 0

⇒ 10x = -26

⇒ x = (-26)/10

⇒ x = -13/5

∴ 10x + 26 = 0 is a rational number.

Given:

Greater number = 120

Calculation:

Let smaller number be x

According to the question

120 – x = x × (50/100)

⇒ 120 – x = x/2

⇒ 240 – 2x = x

⇒ x + 2x = 240

⇒ 3x = 240

⇒ x = 240/3

⇒ x = 80

∴ Sum of both numbers = 120 + 80 = 200

Given:

x² - 7x + 1 = 0

Calculation:

x² - 7x + 1 = 0

Dividing by x:

⇒ x - 7 + 1/x = 0

⇒ x + 1/x = 7

∴ Value of x + 1/x = 7