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Sure! Let's go through each of the questions step by step and solve them:
a) 5 added to a number is 9
To find the number:
1. Let's denote the number by [tex]\( a \)[/tex].
2. According to the statement, [tex]\( a + 5 = 9 \)[/tex].
3. Solving for [tex]\( a \)[/tex] gives us:
[tex]\[ a = 9 - 5 = 4 \][/tex]
So, the number is [tex]\( \boxed{4} \)[/tex].
b) 3 subtracted from a number is equal to 12
To find the number:
1. Let's denote the number by [tex]\( b \)[/tex].
2. According to the statement, [tex]\( b - 3 = 12 \)[/tex].
3. Solving for [tex]\( b \)[/tex] gives us:
[tex]\[ b = 12 + 3 = 15 \][/tex]
So, the number is [tex]\( \boxed{15} \)[/tex].
c) 5 times a number decreased by 2 is 4
To find the number:
1. Let's denote the number by [tex]\( c \)[/tex].
2. According to the statement, [tex]\( 5c - 2 = 4 \)[/tex].
3. Solving for [tex]\( c \)[/tex] gives us:
[tex]\[ 5c = 4 + 2 = 6 \][/tex]
[tex]\[ c = \frac{6}{5} = 1.2 \][/tex]
So, the number is [tex]\( \boxed{1.2} \)[/tex].
d) 2 times the sum of the number [tex]\( x \)[/tex] and 7 is 13
To find the number:
1. Let's denote the number by [tex]\( d \)[/tex].
2. According to the statement, [tex]\( 2(d + 7) = 13 \)[/tex].
3. Solving for [tex]\( d \)[/tex] gives us:
[tex]\[ d + 7 = \frac{13}{2} = 6.5 \][/tex]
[tex]\[ d = 6.5 - 7 = -0.5 \][/tex]
So, the number is [tex]\( \boxed{-0.5} \)[/tex].
i. A number is 12 more than the other. Find the numbers if their sum is 48
To find the numbers:
1. Let's denote the numbers by [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
Since [tex]\( x \)[/tex] is 12 more than [tex]\( y \)[/tex], we have [tex]\( x = y + 12 \)[/tex].
2. According to the statement, their sum is [tex]\( x + y = 48 \)[/tex].
3. Substituting [tex]\( y + 12 \)[/tex] for [tex]\( x \)[/tex] gives us:
[tex]\[ (y + 12) + y = 48 \][/tex]
[tex]\[ 2y + 12 = 48 \][/tex]
[tex]\[ 2y = 48 - 12 = 36 \][/tex]
[tex]\[ y = \frac{36}{2} = 18 \][/tex]
4. Therefore, [tex]\( x = y + 12 = 18 + 12 = 30 \)[/tex].
So, the numbers are [tex]\( \boxed{30} \)[/tex] and [tex]\( \boxed{18} \)[/tex].
ii. Twice the number decreased by 22 is 48 find the number
To find the number:
1. Let's denote the number by [tex]\( n \)[/tex].
2. According to the statement, [tex]\( 2n - 22 = 48 \)[/tex].
3. Solving for [tex]\( n \)[/tex] gives us:
[tex]\[ 2n = 48 + 22 = 70 \][/tex]
[tex]\[ n = \frac{70}{2} = 35 \][/tex]
So, the number is [tex]\( \boxed{35} \)[/tex].
v. [tex]\(\frac{4}{5}\)[/tex] of a number is more than [tex]\(\frac{3}{4}\)[/tex] of the number by 5. Find the number
To find the number:
1. Let's denote the number by [tex]\( m \)[/tex].
2. According to the statement:
[tex]\[ \frac{4}{5}m = \frac{3}{4}m + 5 \][/tex]
3. Solving for [tex]\( m \)[/tex] gives us:
[tex]\[ \frac{4}{5}m - \frac{3}{4}m = 5 \][/tex]
Find a common denominator for the fractions, which is 20:
[tex]\[ \frac{16m}{20} - \frac{15m}{20} = 5 \][/tex]
[tex]\[ \frac{1m}{20} = 5 \][/tex]
[tex]\[ m = 5 \times 20 = 100 \][/tex]
So, the number is [tex]\( \boxed{100} \)[/tex].
vi. In 25 Minutes, a train travels 20 km. How far will it travel in 5 minutes?
To find the distance:
1. Let the distance traveled in 5 minutes be [tex]\( d \)[/tex].
2. According to the statement, the speed of the train is [tex]\( \frac{20 \text{ km}}{25 \text{ min}} = 0.8 \text{ km/min} \)[/tex].
3. Multiply the speed by 5 minutes:
[tex]\[ d = 0.8 \text{ km/min} \times 5 \text{ min} = 4 \text{ km} \][/tex]
So, the train will travel [tex]\( \boxed{4} \)[/tex] km in 5 minutes.
vii. What should be added to [tex]\(4x^3 - 10x^2 + 12x + 6\)[/tex] so that it becomes exactly divisible by [tex]\(2x + 1\)[/tex]?
To find the term that should be added:
1. First, perform polynomial division of [tex]\( 4x^3 - 10x^2 + 12x + 6 \)[/tex] by [tex]\( 2x + 1 \)[/tex].
2. The remainder of this division will be the term that needs to be added to the polynomial.
3. The remainder when [tex]\( 4x^3 - 10x^2 + 12x + 6 \)[/tex] is divided by [tex]\( 2x + 1 \)[/tex] is determined to be [tex]\( 3 \)[/tex].
Therefore, the term that should be added is [tex]\( \boxed{3} \)[/tex].
a) 5 added to a number is 9
To find the number:
1. Let's denote the number by [tex]\( a \)[/tex].
2. According to the statement, [tex]\( a + 5 = 9 \)[/tex].
3. Solving for [tex]\( a \)[/tex] gives us:
[tex]\[ a = 9 - 5 = 4 \][/tex]
So, the number is [tex]\( \boxed{4} \)[/tex].
b) 3 subtracted from a number is equal to 12
To find the number:
1. Let's denote the number by [tex]\( b \)[/tex].
2. According to the statement, [tex]\( b - 3 = 12 \)[/tex].
3. Solving for [tex]\( b \)[/tex] gives us:
[tex]\[ b = 12 + 3 = 15 \][/tex]
So, the number is [tex]\( \boxed{15} \)[/tex].
c) 5 times a number decreased by 2 is 4
To find the number:
1. Let's denote the number by [tex]\( c \)[/tex].
2. According to the statement, [tex]\( 5c - 2 = 4 \)[/tex].
3. Solving for [tex]\( c \)[/tex] gives us:
[tex]\[ 5c = 4 + 2 = 6 \][/tex]
[tex]\[ c = \frac{6}{5} = 1.2 \][/tex]
So, the number is [tex]\( \boxed{1.2} \)[/tex].
d) 2 times the sum of the number [tex]\( x \)[/tex] and 7 is 13
To find the number:
1. Let's denote the number by [tex]\( d \)[/tex].
2. According to the statement, [tex]\( 2(d + 7) = 13 \)[/tex].
3. Solving for [tex]\( d \)[/tex] gives us:
[tex]\[ d + 7 = \frac{13}{2} = 6.5 \][/tex]
[tex]\[ d = 6.5 - 7 = -0.5 \][/tex]
So, the number is [tex]\( \boxed{-0.5} \)[/tex].
i. A number is 12 more than the other. Find the numbers if their sum is 48
To find the numbers:
1. Let's denote the numbers by [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
Since [tex]\( x \)[/tex] is 12 more than [tex]\( y \)[/tex], we have [tex]\( x = y + 12 \)[/tex].
2. According to the statement, their sum is [tex]\( x + y = 48 \)[/tex].
3. Substituting [tex]\( y + 12 \)[/tex] for [tex]\( x \)[/tex] gives us:
[tex]\[ (y + 12) + y = 48 \][/tex]
[tex]\[ 2y + 12 = 48 \][/tex]
[tex]\[ 2y = 48 - 12 = 36 \][/tex]
[tex]\[ y = \frac{36}{2} = 18 \][/tex]
4. Therefore, [tex]\( x = y + 12 = 18 + 12 = 30 \)[/tex].
So, the numbers are [tex]\( \boxed{30} \)[/tex] and [tex]\( \boxed{18} \)[/tex].
ii. Twice the number decreased by 22 is 48 find the number
To find the number:
1. Let's denote the number by [tex]\( n \)[/tex].
2. According to the statement, [tex]\( 2n - 22 = 48 \)[/tex].
3. Solving for [tex]\( n \)[/tex] gives us:
[tex]\[ 2n = 48 + 22 = 70 \][/tex]
[tex]\[ n = \frac{70}{2} = 35 \][/tex]
So, the number is [tex]\( \boxed{35} \)[/tex].
v. [tex]\(\frac{4}{5}\)[/tex] of a number is more than [tex]\(\frac{3}{4}\)[/tex] of the number by 5. Find the number
To find the number:
1. Let's denote the number by [tex]\( m \)[/tex].
2. According to the statement:
[tex]\[ \frac{4}{5}m = \frac{3}{4}m + 5 \][/tex]
3. Solving for [tex]\( m \)[/tex] gives us:
[tex]\[ \frac{4}{5}m - \frac{3}{4}m = 5 \][/tex]
Find a common denominator for the fractions, which is 20:
[tex]\[ \frac{16m}{20} - \frac{15m}{20} = 5 \][/tex]
[tex]\[ \frac{1m}{20} = 5 \][/tex]
[tex]\[ m = 5 \times 20 = 100 \][/tex]
So, the number is [tex]\( \boxed{100} \)[/tex].
vi. In 25 Minutes, a train travels 20 km. How far will it travel in 5 minutes?
To find the distance:
1. Let the distance traveled in 5 minutes be [tex]\( d \)[/tex].
2. According to the statement, the speed of the train is [tex]\( \frac{20 \text{ km}}{25 \text{ min}} = 0.8 \text{ km/min} \)[/tex].
3. Multiply the speed by 5 minutes:
[tex]\[ d = 0.8 \text{ km/min} \times 5 \text{ min} = 4 \text{ km} \][/tex]
So, the train will travel [tex]\( \boxed{4} \)[/tex] km in 5 minutes.
vii. What should be added to [tex]\(4x^3 - 10x^2 + 12x + 6\)[/tex] so that it becomes exactly divisible by [tex]\(2x + 1\)[/tex]?
To find the term that should be added:
1. First, perform polynomial division of [tex]\( 4x^3 - 10x^2 + 12x + 6 \)[/tex] by [tex]\( 2x + 1 \)[/tex].
2. The remainder of this division will be the term that needs to be added to the polynomial.
3. The remainder when [tex]\( 4x^3 - 10x^2 + 12x + 6 \)[/tex] is divided by [tex]\( 2x + 1 \)[/tex] is determined to be [tex]\( 3 \)[/tex].
Therefore, the term that should be added is [tex]\( \boxed{3} \)[/tex].
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