Grade 7-8 Questions on Fractions and decimals Whole numbers Naming large numbers Perimeter and area of shapes Unit conversion

To reflect over the y-axis, negate the x-coordinates: A becomes (-1, 1), B becomes (-4, 1), and C becomes (-4, 4). Then translate each point 3 units up by adding 3 to the y-coordinates: A(-1, 4), B(-4, 4), C(-4, 7).

To find the first term greater than 50, solve the inequality \( 3n + 2 > 50 \). Subtract 2: \( 3n > 48 \). Divide by 3: \( n > 16 \). The smallest integer value of \( n \) is 17. Substituting \( n = 17 \), \( t_{17} = 3(17) + 2 = 51 \).

In Pascal’s Triangle, the sum of the numbers in the \( n \)-th row is \( 2^{n-1} \). For the 7th row, \( n = 7 \), so the sum is \( 2^6 = 64 \).

The prime factorization of 72 is \( 2^3 \times 3^2 \), and for 120 it is \( 2^3 \times 3 \times 5 \). The GCF is the product of the smallest powers of all common primes: \( 2^3 \times 3 = 24 \). The LCM is the product of the highest powers of all primes: \( 2^3 \times 3^2 \times 5 = 360 \).

Simplify each term: \( 2^5 \times 3^3 = 32 \times 27 = 864 \), and \( 2^3 \times 3^2 = 8 \times 9 = 72 \). Subtract: \( 864 - 72 = 792 \).

Convert \( -\frac{5}{8} \) to a decimal: \( -0.625 \). Compare \( -0.625 \) and \( -0.6 \). Since \( -0.6 \) is closer to 0, it is larger.

To reflect over the y-axis, negate the x-coordinates: A becomes (-1, 1), B becomes (-4, 1), and C becomes (-4, 4). Then translate each point 3 units up by adding 3 to the y-coordinates: A(-1, 4), B(-4, 4), C(-4, 7).

To find the first term greater than 50, solve the inequality \( 3n + 2 > 50 \). Subtract 2: \( 3n > 48 \). Divide by 3: \( n > 16 \). The smallest integer value of \( n \) is 17. Substituting \( n = 17 \), \( t_{17} = 3(17) + 2 = 51 \).

In Pascal’s Triangle, the sum of the numbers in the \( n \)-th row is \( 2^{n-1} \). For the 7th row, \( n = 7 \), so the sum is \( 2^6 = 64 \).

The prime factorization of 72 is \( 2^3 \times 3^2 \), and for 120 it is \( 2^3 \times 3 \times 5 \). The GCF is the product of the smallest powers of all common primes: \( 2^3 \times 3 = 24 \). The LCM is the product of the highest powers of all primes: \( 2^3 \times 3^2 \times 5 = 360 \).

Simplify each term: \( 2^5 \times 3^3 = 32 \times 27 = 864 \), and \( 2^3 \times 3^2 = 8 \times 9 = 72 \). Subtract: \( 864 - 72 = 792 \).

Convert \( -\frac{5}{8} \) to a decimal: \( -0.625 \). Compare \( -0.625 \) and \( -0.6 \). Since \( -0.6 \) is closer to 0, it is larger.

The surface area of a rectangular prism is given by \( 2(lw + lh + wh) \). Substituting the dimensions: \( 2(8 \times 5 + 8 \times 3 + 5 \times 3) = 2(40 + 24 + 15) = 2(79) = 158 \) square cm.

The speed of the car is \( \frac{240}{3} = 80 \) km/h. To find the time for 400 km, divide the distance by the speed: \( \frac{400}{80} = 5 \) hours.

Distribute the 2: \( 3x + 8 - 2x \). Combine like terms: \( (3x - 2x) + 8 = x + 8 \).

Using substitution, solve the second equation for \( x \): \( x = y + 1 \). Substitute into the first equation: \( 2(y + 1) + y = 7 \). Simplify: \( 2y + 2 + y = 7 \). Combine like terms: \( 3y + 2 = 7 \). Solve: \( y = 5/3 \). Then find \( x \): \( x = 5/3 + 1 = 8/3 \).

The volume of a cylinder is \( \pi r^2 h \). Substituting the values: \( 3.14 \times 5^2 \times 10 = 3.14 \times 25 \times 10 = 785 \) cm³.

Let the width be \( w \). Then the length is \( 2w \). The perimeter is \( 2(w + 2w) = 36 \). Simplify: \( 6w = 36 \). Solve for \( w \): \( w = 6 \). The length is \( 2(6) = 12 \). The dimensions are 6 cm by 12 cm.

Move the decimal point 4 places to the right to get \( 5.6 \). Multiply by \( 10^{-4} \) to account for the shift: \( 5.6 \times 10^{-4} \).

The possible outcomes for two dice are 6 × 6 = 36. To get a sum of 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1), there are 6 outcomes. The probability is \( \frac{6}{36} = \frac{1}{6} \).

Expand the left-hand side: \( 3x - 12 = 2x + 5 \). Subtract \( 2x \) from both sides: \( x - 12 = 5 \). Add 12 to both sides: \( x = 17 \).

Let the integers be \( x \), \( x+1 \), and \( x+2 \). Their sum is \( x + (x+1) + (x+2) = 72 \). Simplify: \( 3x + 3 = 72 \). Subtract 3: \( 3x = 69 \). Divide by 3: \( x = 23 \). The integers are 23, 24, and 25.

Find a common denominator: the least common multiple of 3 and 6 is 6. Rewrite the fractions: \( \frac{2}{3} = \frac{4}{6} \). Add: \( \frac{4}{6} + \frac{5}{6} = \frac{9}{6} \). Simplify: \( \frac{9}{6} = \frac{3}{2} \).

Use the Pythagorean theorem: \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse. Check: \( 7^2 + 24^2 = 49 + 576 = 625 \), and \( 25^2 = 625 \). Since both sides are equal, it is a right triangle.

Add the two equations: \( (x + y) + (x - y) = 10 + 4 \), which simplifies to \( 2x = 14 \). Solve for \( x \): \( x = 7 \). Substitute into the first equation: \( 7 + y = 10 \). Solve for \( y \): \( y = 3 \).

Recognize that \( x^2 - 9 \) is a difference of squares: \( x^2 - 9 = (x - 3)(x + 3) \).

Calculate the discount: \( 15% \times 80 = 0.15 \times 80 = 12 \). Subtract the discount from the original price: \( 80 - 12 = 68 \). The final price is $68.