Arithmetic and Geometry Practice

If 0.6a = 1.2b, then a = 2b. Substituting a = 2b into the expression 2a - 4b gives 2(2b) - 4b, which is 4b - 4b = 0. Therefore, the result is always zero regardless of the values of a and b.

Weight scales with volume, and volume of a sphere is proportional to the cube of its radius. If the radius is multiplied by 3, the volume increases by 3^3 = 27 times. Therefore, a 12-pound block would weigh 12 × 27 = 324 pounds with 3 times the radius.

For any angle A, sin²(A) + cos²(A) = 1, so cos(A) = √(1 - sin²(A)). Given sin(A) = 11/14, we have cos(A) = √(1 - 121/196) = √(75/196) = √75/14 = 5√3/14. The cosine is positive since A is acute.

To find the average, add the fractions and divide by 2. 7/8 + 3/4 = 7/8 + 6/8 = 13/8. Dividing 13/8 by 2 gives 13/16. The other options result from incorrect common denominators or failing to divide by two.

To find 5% of 6%, convert percentages to decimals: 5% = 0.05 and 6% = 0.06. Multiply: 0.05 × 0.06 = 0.0030. The decimal 0.0030 equals 0.3%, but as a pure decimal it is 0.003. The options appear to be in decimal form, so 0.003 is correct, which matches option C.

After 3/8 take French, 5/8 remain. Of these remaining, 1/5 take German, so 4/5 of the remaining students take Italian. Therefore, Italian students = (4/5) × (5/8) = 4/8 = 1/2 = 50%. The key is recognizing that 4/5 of the remaining students (not 1/5) study Italian.

L1 has slope (3-2)/(2-0) = 1/2. Since L1 and L2 are perpendicular, L2's slope must be -2 (negative reciprocal). Using point-slope form through (2,3): y-3 = -2(x-2). Checking point (3,1): 1-3 = -2 and -2(3-2) = -2, so it satisfies the equation. The other options do not satisfy this line equation.

Subtracting 123456 from 654321 gives 530865. This is a straightforward arithmetic calculation: 654321 - 123456 = 530865. You can verify by adding 530865 to 123456, which returns 654321.

The product contains 10 as a factor, which guarantees the last digit is 0. When you multiply any integer by 10, the result ends in 0. This is a pattern recognition question - you don't need to calculate the full product.

The cube root of 60698457 is 393, since 393 × 393 × 393 = 60698457. This is a numerical computation question requiring you to identify which option cubed equals the given number. Note the grammar issue in the question.

To convert bytes to gigabytes, divide by 1 billion (1 GB = 1,000,000,000 bytes in decimal system). 170,000,000 bytes ÷ 1,000,000,000 = 0.17 GB. Option A (1.7) would be correct if converting to megabytes, while options B and C are miscalculations.

Multiply 1111 by 111 using the distributive property: 1111 × 111 = 1111 × (100 + 10 + 1) = 111,100 + 11,110 + 1,111 = 123,321. This can also be verified by direct multiplication or using the pattern that 1111 × 111 produces a palindromic result.