Simplification Questions and Answers

Answer: A

The value is 39. Square each decimal: 4.5 x 4.5 = 20.25 and 3.5 x 3.5 = 12.25. Adding the last term gives 20.25 + 12.25 + 6.5 = 39. Options 32, 29 and 38 would come from mistakes such as dropping a part of the product or the final 6.5. A quick check: 4.5 squared is (4 + 0.5) squared = 16 + 4 + 0.25 = 20.25, and 3.5 squared is (3 + 0.5) squared = 9 + 3 + 0.25 = 12.25. The total 20.25 + 12.25 is exactly 32.5, which with the 6.5 makes 39.

Answer: B

The sum is 49. There are 98 terms, each of the form 1 minus k/99. Their sum is 98 minus (1 + 2 + ... + 98)/99. The sum of 1 to 98 is 98 x 99 / 2 = 4851, and 4851/99 = 49. So the total is 98 - 49 = 49. Option 98 forgets to subtract the fractions, and 99 and 50 do not match either the term count or the subtracted total. As a check, pair the first and the last terms: (1 - 1/99) + (1 - 98/99) = 2 - 99/99 = 1. There are 49 such pairs among the 98 terms, so the sum is 49 x 1 = 49, agreeing with the main working.

Answer: D

Round each number to the nearest convenient value. 5.01 x 19.99 is about 5 x 20 = 100. 99.99 / 25.01 is about 100 / 25 = 4. 34.11 is about 34. So the equation becomes 100 + 4 - 34 - ? = 0, which gives ? = 70. The answer is 70, option (D). The options 60, 80 and 50 would come from a mistake in the first product or in the sign of the last term, and the exact value, 70.0, is close to 70.

Answer: A

Use the identity for the difference of two squares, which says that a squared minus b squared equals (a + b) times (a - b).

\[ 39^{2} - 38^{2} = (39 + 38)(39 - 38) \]

\[ = 77 \times 1 = 77 \]

So the value is 77, option (A). A check: 39 squared is 1521 and 38 squared is 1444, and their difference is 77. The values 108, 117 and 57 come from adding or subtracting the numbers instead of using the identity correctly. The identity is also used to calculate in the head: any difference of the squares of two consecutive numbers is the sum of the two numbers.

Answer: D

Work out the top and bottom of the fraction separately, using the order of operations. The top is 24 x 13 - 16 + 84 / (6 x 2) = 312 - 16 + 7 = 303. The bottom is 28 / 4 + 48 x (16 / 96) = 7 + 8 = 15.

\[ \frac{303}{15} = \frac{101}{5} \]

So the value is 101/5, option (D). The value 302/15 comes from an error of one in the top, and 5 and 91/3 do not follow from the two parts, since 5 would need a top of 75 and 91/3 would need a different bottom.

Answer: B

The expression is the mixed number P and 1/3, plus 4 and 1/R, equal to 9 and 5/6. Take the whole numbers and fractions separately: the fractions are 1/3 and 1/R, and the whole numbers are P and 4. Trying P = 5 gives 5 + 4 = 9, and the fractions must then make up 5/6, which needs 1/3 + 1/R = 5/6, so 1/R = 1/2 and R = 2. Then P - R = 5 - 2 = 3. So the difference is 3, option (B). A different value of P cannot make the fractions fit with a whole number R.

Answer: C

Write each factor as a fraction: 1 + 1/2 = 3/2, 1 + 1/3 = 4/3, 1 + 1/4 = 5/4, and so on up to 1 + 1/24 = 25/24. The product is

\[ \frac{3}{2}\times\frac{4}{3}\times\frac{5}{4}\times\cdots \]

\[ \cdots\times\frac{25}{24} \]

Each numerator cancels the next denominator, so only the 25 at the end and the 2 at the beginning are left, giving 25/2 = 12.5. So the value is 12.5, option (C). The options 0.5, 10 and 25 do not come from the telescoping product.

Answer: B

The approximate value is 217. Following BODMAS, multiplication and division come before addition and subtraction: 37.39 × 50.89 ≈ 1903, and 1903 ÷ 17.17 ≈ 110.8. Then 129.78 + 110.8 = 240.6, and 240.6 − 23.69 ≈ 216.9, which is nearest to 217.

\[ \frac{37.39 \times 50.89}{17.17} \approx 110.8 \]

\[ 129.78 + 110.8 - 23.69 \approx 217 \]

The options 289, 127 and 345 are far from this value, so the second option is the answer.

Answer: C

The value is 19/20. Using BODMAS, 'of' is done first, then division and multiplication, then addition and subtraction. The first part is the division and multiplication:

\[ \frac{21}{4} \div \frac{7}{2} \times \frac{5}{6} \]

\[ = \frac{21}{4} \times \frac{2}{7} \times \frac{5}{6} \]

\[ = \frac{5}{4} \]

Next, the 'of' term and the divisions that follow it:

\[ \frac{2}{15} \text{ of } 1\frac{3}{5} \]

\[ = \frac{2}{15} \times \frac{8}{5} = \frac{16}{75} \]

\[ \frac{2}{5} \div \frac{16}{75} = \frac{15}{8} \]

\[ \frac{3}{4} \div \frac{5}{12} = \frac{9}{5} \]

Combining all the terms:

\[ \frac{5}{4} - \frac{3}{8} + \frac{15}{8} - \frac{9}{5} \]

\[ = \frac{11}{4} - \frac{9}{5} = \frac{19}{20} \]

The other options, 9/10, 3/5 and 39/40, come from a wrong order of operations.

Answer: A

The approximate value is 257. Round each number: 45.67 ÷ 22.89 is about 2, 56.69 is about 57, and 23.89 × 13.37 is about 24 × 13. \[ 2 - 57 + 24 \times 13 \] \[ 2 - 57 + 312 = 257 \] Exact working gives a value near 265, and among the options only 257 is anywhere close, because 216, 345 and 378 are all far from it. The question asks for an approximate value, so the option nearest the estimate is correct. A second look at the exact figures shows that 45.67 ÷ 22.89 is just under 2, 23.89 × 13.37 is about 319 and the sum is about 265, which still leaves 257 as the closest option.