1. **Problem:** The price of 3 soap bars is 420. Find the price of 5 soap bars. **Step 1:** Find the price of 1 soap bar by dividing 420 by 3. $$ \text{Price of 1 soap bar} = \frac{420}{3} = 140 $$ **Step 2:** Find the price of 5 soap bars by multiplying the price of 1 soap bar by 5. $$ \text{Price of 5 soap bars} = 140 \times 5 = 700 $$ **Answer:** Price of 5 soap bars is 700. 2. **Problem:** If the general term of a number pattern is $5 (12 - n)$, find the second term. **Step 1:** Substitute $n=2$ into the general term. $$ 5(12 - 2) = 5 \times 10 = 50 $$ **Answer:** The second term is 50. 3. **Problem:** Factorize $x^2 + 5x - 6$. **Step 1:** Find two numbers that multiply to $-6$ and add to $5$. These numbers are $6$ and $-1$. **Step 2:** Write as $$ x^2 + 6x - x - 6 $$ **Step 3:** Factor by grouping: $$ x(x+6) - 1(x+6) = (x - 1)(x + 6) $$ **Answer:** Factorized form is $(x - 1)(x + 6)$. 4. **Problem:** Find the value of $15x - 8y$ if $x=\frac{3}{5}$ and $y=\frac{1}{4}$. **Step 1:** Substitute values: $$ 15 \times \frac{3}{5} - 8 \times \frac{1}{4} = 9 - 2 = 7 $$ **Answer:** The value is 7. 5. **Problem:** Simplify $$ \frac{3a}{5a + 4} - \frac{a + 1}{5a + 4} $$. **Step 1:** Because both fractions share the denominator $5a + 4$, combine numerators: $$ \frac{3a - (a + 1)}{5a + 4} = \frac{3a - a - 1}{5a + 4} = \frac{2a - 1}{5a + 4} $$ **Answer:** Simplified form is $$\frac{2a - 1}{5a + 4}$$. 6. **Problem:** Round 879.645 to: (i) First decimal place (ii) Whole number **(i) Step 1:** Look at second decimal place, which is 4 (<5), so the first decimal stays 6. Rounded to first decimal place: 879.6 **(ii) Step 1:** Look at first decimal place, which is 6 (≥5), so round up the whole number. Rounded to whole number: 880 **Answer:** (i) 879.6, (ii) 880 7. **Problem:** Given $a^3 \times a^{\Box} / a^3 = a^{\Box} = \frac{1}{a^4}$, find the suitable values in blanks. **Step 1:** Simplify left side: $$ \frac{a^3 \times a^x}{a^3} = a^x $$ Given that equals $\frac{1}{a^4} = a^{-4}$ So, $x = -4$ **Answer:** Values are $x = -4$. 8. **Problem:** In triangle ABC, if $\hat{A} + \hat{B} = 130^\circ$ and $\hat{A} + \hat{C} = 85^\circ$, find $\hat{A}$. **Step 1:** Use the triangle angle sum property: $\hat{A} + \hat{B} + \hat{C} = 180^\circ$. **Step 2:** From $\hat{A} + \hat{B} = 130^\circ$, we get $\hat{B} = 130^\circ - \hat{A}$. From $\hat{A} + \hat{C} = 85^\circ$, we get $\hat{C} = 85^\circ - \hat{A}$. **Step 3:** Substitute in triangle sum: $$ \hat{A} + (130 - \hat{A}) + (85 - \hat{A}) = 180 $$ Simplify: $$ \hat{A} + 130 - \hat{A} + 85 - \hat{A} = 180 $$ $$ 215 - \hat{A} = 180 $$ $$ -\hat{A} = -35 $$ $$ \hat{A} = 35^\circ $$ **Answer:** $35^\circ$ 9. **Problem:** Evaluate $1101_2 + 110_2$. **Step 1:** Convert binary to decimal. $$1101_2 = 1\times2^3 + 1\times2^2 + 0 + 1 = 8 + 4 + 0 + 1 = 13$$ $$110_2 = 1\times2^2 + 1\times2^1 + 0 = 4 + 2 + 0 = 6$$ **Step 2:** Add decimals: $13 + 6 = 19$ **Step 3:** Convert back to binary: 19 decimal is $10011_2$ **Answer:** $10011_2$ 10. **Problem:** Write the elements of $B'$ (complement of set B) given the sets in the diagram. **Step 1:** Universal set contains numbers: 1,2,3,4,5,6,7,8,9. **Step 2:** Set B contains: 4,5,6,7 **Step 3:** Elements not in B are: $$ B' = \{1,2,3,8,9\} $$ **Answer:** $B' = \{1,2,3,8,9\}$ 11. **Problem:** For $PT - AB = AP$, make $P$ the subject. **Step 1:** The original equation is $PT - AB = AP$ **Step 2:** Rearrange: $PT - AP = AB$ **Step 3:** Factor out $P$: $P(T - A) = AB$ **Step 4:** Divide both sides by $(T - A)$: $$ P = \frac{AB}{T - A} $$ **Answer:** $P = \frac{AB}{T - A}$ 12. **Problem:** Area of the bottom of a cuboid shaped tank is 4500 cm². It is filled with water up to 60 cm high. Find volume of water in litres. **Step 1:** Volume $= \text{Area} \times \text{Height}$ $$ V = 4500 \times 60 = 270000 \text{ cm}^3 $$ **Step 2:** Convert $cm^3$ to litres (1000 $cm^3$ = 1 litre): $$ \frac{270000}{1000} = 270 \text{ litres} $$ **Answer:** Volume of water is 270 litres.