Q1. Compute \( \frac{5}{12}\times \frac{7}{18} \) by cancelling common factors.

Q2. Simplify \( \frac{12}{7}\times \frac{5}{24} \) to lowest terms.

Cancel \(12\) with \(24\): \( \frac{1}{7}\times\frac{5}{2}=\frac{5}{14} \).

Q3. Evaluate \( \frac{3}{5}\times 4 \) and write as mixed number.

\( \frac{3}{5}\times \frac{4}{1}=\frac{12}{5}=2\frac{2}{5} \).

Q4. Compute \( 4\times \frac{1}{3} \) and interpret.

\( \frac{4}{3}=1\frac{1}{3} \). It is one and one-third.

Q5. Divide \( \frac{2}{3}\div \frac{3}{5} \).

Multiply by reciprocal: \( \frac{2}{3}\times \frac{5}{3}=\frac{10}{9}=1\frac{1}{9} \).

Q6. Divide \( \frac{1}{5}\div \frac{1}{2} \).

\( \frac{1}{5}\times \frac{2}{1}=\frac{2}{5} \).

Q7. Prove \( \frac{1}{b}\times \frac{1}{d}=\frac{1}{bd} \) by area model.

Q8. Which is bigger: \( \frac{12}{15}\) of \(500\) g or \( \frac{3}{20}\) of \(4\) kg?

Q9. If a tap fills \( \frac{7}{10}\) tank in 1 hour, how much in \( \frac{3}{4}\) hr?

\( \frac{3}{4}\times \frac{7}{10}=\frac{21}{40} \) of tank.

Q10. Show \( a\div \frac{1}{n}=a\times n \) with \(a=\frac{p}{q}\).

\( \frac{p}{q}\div \frac{1}{n}=\frac{p}{q}\times \frac{n}{1}=\frac{pn}{q}=a\times n \).

Q11. Convert and multiply: \( 1\frac{1}{4}\times 8 \).

\( \frac{5}{4}\times 8=\frac{5}{4}\times \frac{8}{1}=10 \).

Q12. Distance: \( \frac{2}{5}\) hr at \(3\) km/h.

Distance \(=\frac{2}{5}\times 3=\frac{6}{5}=1.2\) km.

Q13. Multiply \( \frac{5}{4}\times \frac{3}{2} \).

\( \frac{15}{8}=1\frac{7}{8} \).

Q14. Compute \( \left(\frac{3}{4}\right)\div 3 \).

\( \frac{3}{4}\times \frac{1}{3}=\frac{1}{4} \).

Q15. Area: \( 3\frac{3}{4}\) ft by \( 9\frac{3}{5}\) ft.

Q16. \( \frac{4}{6}\times \frac{3}{5} \) in lowest terms.

\( \frac{2}{3}\times \frac{3}{5}=\frac{2}{5} \).

Q17. Compare \( \frac{1}{4}\times 8 \) to 8.

Product \(=2\), which is less than 8.

Q18. If \(x\times \frac{3}{5}= \frac{9}{10}\), find \(x\).

\( x=\frac{9}{10}\div \frac{3}{5}=\frac{9}{10}\times \frac{5}{3}=\frac{3}{2}=1\frac12 \).

Q19. \( \left(\frac{2}{3}\right)^2\times \left(\frac{3}{2}\right) \)

\( \frac{4}{9}\times \frac{3}{2}=\frac{12}{18}=\frac{2}{3} \).

Q20. Show \( \frac{a}{b}\div \frac{c}{d}=\frac{ad}{bc} \).

By reciprocal: \( \frac{a}{b}\times \frac{d}{c}=\frac{ad}{bc} \).

Q1. A farmer has 5 grandchildren. She gives each \( \frac{2}{3}\) acre. How much in total?

• Total \(=5\times \frac{2}{3}=\frac{10}{3}=3\frac{1}{3}\) acres.
• Method: repeated addition \( \frac{2}{3}\) five times.

Q2. Internet costs ₹8 per hour. Find the cost of \(1\frac{1}{4}\) hours.

• \(1\frac{1}{4}=\frac{5}{4}\) hr.
• Cost \(=\frac{5}{4}\times 8=10\). So ₹10.

Q3. Tenzin drinks \( \frac12\) glass milk daily. How many glasses in a week & in January?

• Week (7 days): \(7\times \frac12=\frac{7}{2}=3\frac12\) glasses.
• January (31 days): \(31\times \frac12=\frac{31}{2}=15\frac12\) glasses.

Q4. Work team makes 1 km canal in 8 days. Per day? Per week (5 days)?

• Per day \(=\frac{1}{8}\) km.
• Per 5-day week \(=5\times \frac{1}{8}=\frac{5}{8}\) km.

Q5. 5 L oil shared equally among 3 families weekly. Per family per week? In 4 weeks?

• Each week per family \(=5\div 3=\frac{5}{3}=1\frac{2}{3}\) L.
• In 4 weeks \(=4\times \frac{5}{3}=\frac{20}{3}=6\frac{2}{3}\) L.

Q6. Moon sets \( \frac{5}{6}\) hr later daily. From Mon 10 pm, when on Thu?

• Total delay by Thu \(=3\times \frac{5}{6}=\frac{15}{6}=\frac{5}{2}=2.5\) hr.
• 10:00 pm \(+\) 2 h 30 m = 12:30 am (Thursday).

Q7. Multiply and convert to mixed: (a) \(7\times \frac{3}{5}\), (b) \(4\times \frac{1}{3}\)

• (a) \( \frac{21}{5}=4\frac{1}{5}\). (b) \( \frac{4}{3}=1\frac{1}{3}\).

Q8. Multiply and convert: (c) \( \frac{9}{7}\times 6\), (d) \( \frac{13}{11}\times 6\)

• (c) \( \frac{9}{7}\times \frac{6}{1}=\frac{54}{7}=7\frac{5}{7}\).
• (d) \( \frac{13}{11}\times 6=\frac{78}{11}=7\frac{1}{11}\).

Q9. Two-fraction product by area model: \( \frac{1}{3}\times \frac{1}{5} \) etc.

• (a) \( \frac{1}{15}\), (b) \( \frac{1}{12}\), (c) \( \frac{1}{10}\), (d) \( \frac{1}{30}\).
• \( \frac{1}{12}\times \frac{1}{18}=\frac{1}{216}\).

Q10. Compute: (a) \( \frac{2}{3}\times \frac{4}{5}\), (b) \( \frac14\times \frac23\), (c) \( \frac35\times \frac12\), (d) \( \frac46\times \frac35\)

• (a) \( \frac{8}{15}\). (b) \( \frac{2}{12}=\frac{1}{6}\).
• (c) \( \frac{3}{10}\). (d) \( \frac{4}{6}\times \frac{3}{5}=\frac{2}{3}\times \frac{3}{5}=\frac{2}{5}\).

Q11. Water tank fills \( \frac{7}{10}\) in 1 hr. Find filled part for: (a) \( \frac13\) hr, (b) \( \frac23\) hr, (c) \( \frac34\) hr, (d) \( \frac{7}{10}\) hr. Also time for full tank.

• (a) \( \frac13\times \frac{7}{10}=\frac{7}{30}\).
• (b) \( \frac23\times \frac{7}{10}=\frac{14}{30}=\frac{7}{15}\).
• (c) \( \frac34\times \frac{7}{10}=\frac{21}{40}\).
• (d) \( \frac{7}{10}\times \frac{7}{10}=\frac{49}{100}\).
• For full tank: rate \(=\frac{7}{10}\) per hr ⇒ time \(= \frac{1}{\frac{7}{10}}=\frac{10}{7}\) hr \(=1\) hr \(+\) \( \frac{3}{7}\) hr ≈ 1 h 25 min 43 s.

Q12. Land: Government takes \( \frac{1}{6}\). Remaining shared: half to Krishna, one-third to Bora, rest with Somu. Find each as part of original.

• Remaining \(=1-\frac{1}{6}=\frac{5}{6}\).
• Krishna \(=\frac{1}{2}\times \frac{5}{6}=\frac{5}{12}\).
• Bora \(=\frac{1}{3}\times \frac{5}{6}=\frac{5}{18}\).
• Somu \(=\frac{5}{6}-\left(\frac{5}{12}+\frac{5}{18}\right)=\frac{30-15-10}{36}=\frac{5}{36}\).

Q13. Four saplings in a row, distance between neighbors \( \frac{3}{4}\) m. Distance first to last?

There are 3 gaps: \( 3\times \frac{3}{4}=\frac{9}{4}=2\frac{1}{4}\) m.

Q14. “Is product always greater?” Decide for: \( \frac{1}{4}\times 8\) and \( \frac{3}{4}\times \frac{2}{5}\).

• \( \frac{1}{4}\times 8=2\), which is \(<8\) but \(> \frac14\).
• \( \frac{3}{4}\times \frac{2}{5}=\frac{3}{10}\), less than both factors (both <1).

Q15. Why does \( \frac{a}{b}\times \frac{c}{d}=\frac{ac}{bd}\) also show commutativity?

Because \(ac=ca\) and \(bd=db\), swapping order doesn’t change the product.

Q16. Leena used \( \frac14\) L milk for 5 cups. Milk per cup?

\( \frac{1}{4}\div 5=\frac{1}{4}\times \frac{1}{5}=\frac{1}{20}\) L per cup.

Q17. Bricks: Cover \( 7\frac12\) sq units with squares of side \( \frac{1}{5}\) units. How many?

• Area of 1 brick \(=\frac{1}{5}\times \frac{1}{5}=\frac{1}{25}\).
• Total area \(= \frac{15}{2}\).
• Number \(= \frac{15}{2}\div \frac{1}{25}=\frac{375}{2}=187\frac12\). Need 188 whole bricks to cover fully.

Q18. Fountains: 1 day, \( \frac12\) day, \( \frac14\) day, \( \frac15\) day. Time if all open?

• Rates/day: \(1,\ 2,\ 4,\ 5\). Sum \(=12\).
• Time \(=\frac{1}{12}\) day = 2 hours.

Q19. “Dramma-tic” gift: \( \frac12\times \frac23\times \frac34\times \frac{1}{5}\times \frac{1}{16}\times \frac{1}{4}\) of a dramma. How many cowries if 1 dramma = 1280 cowries?

• Product \(=\frac{6}{7680}=\frac{1}{1280}\).
• Cowries \(= 1280\times \frac{1}{1280}=1\) cowrie.

Q20. Ant colony splits equally at each fork and reaches two sources. What fraction reaches each?

At each split, fraction halves. After \(n\) equal splits along a route, the portion along that route is \( \frac{1}{2^n}\). For a simple single path with \(n\) splits to each source, each source gets \( \frac{1}{2^n}\) of the original group.

Exercise 1(a–d). Find the products using a unit square idea: (a) \( \frac{1}{3}\times \frac{1}{5}\), (b) \( \frac{1}{4}\times \frac{1}{3}\), (c) \( \frac{1}{5}\times \frac{1}{2}\), (d) \( \frac{1}{6}\times \frac{1}{5}\). Then find \( \frac{1}{12}\times \frac{1}{18}\).

• (a) \( \frac{1}{15}\), (b) \( \frac{1}{12}\), (c) \( \frac{1}{10}\), (d) \( \frac{1}{30}\).
• \( \frac{1}{12}\times \frac{1}{18}=\frac{1}{216}\).

Exercise 2(a–d). Compute: (a) \( \frac{2}{3}\times \frac{4}{5}\), (b) \( \frac{1}{4}\times \frac{2}{3}\), (c) \( \frac{3}{5}\times \frac{1}{2}\), (d) \( \frac{4}{6}\times \frac{3}{5}\).

• (a) \( \frac{8}{15}\), (b) \( \frac{1}{6}\), (c) \( \frac{3}{10}\), (d) \( \frac{2}{5}\).

Exercise 3. Water tank fills \( \frac{7}{10}\) in 1 hr. How much fills if tap is open for (a) \( \frac{1}{3}\) hr, (b) \( \frac{2}{3}\) hr, (c) \( \frac{3}{4}\) hr, (d) \( \frac{7}{10}\) hr, and (e) time for full tank?

• (a) \( \frac{7}{30}\) ; (b) \( \frac{7}{15}\) ; (c) \( \frac{21}{40}\) ; (d) \( \frac{49}{100}\).
• (e) \( \tfrac{10}{7}\) hr \(=1\) hr \(25\) min \(43\) s (approx).

Exercise 4. Land sharing (Somu, Krishna, Bora) as described: find each share of original land.

• Remaining after road \(=\frac{5}{6}\).
• Krishna \(=\frac{5}{12}\), Bora \(=\frac{5}{18}\), Somu \(=\frac{5}{36}\).

Exercise 5. Area of rectangle \( 3\frac{3}{4}\) ft by \( 9\frac{3}{5}\) ft.

\( \frac{15}{4}\times \frac{48}{5}=36\ \text{sq ft} \).

Exercise 6. Four saplings with \( \frac{3}{4}\) m between neighbors. Distance first to last.

\(3\times \frac{3}{4}=\frac{9}{4}=2\frac{1}{4}\) m.

Exercise 7. Heavier: \( \frac{12}{15}\) of \(500\) g or \( \frac{3}{20}\) of \(4\) kg?

\(400\) g vs \(600\) g ⇒ \( \frac{3}{20}\) of 4 kg is heavier.

Exercise 8. Division patterns: When is quotient < dividend or > dividend?

If divisor \(>1\), quotient \(<\) dividend. If divisor between \(0\) and \(1\), quotient \(>\) dividend.

Exercise 9. Leena’s tea: \( \frac14\) L for 5 cups → milk per cup?

\( \frac{1}{4}\div 5=\frac{1}{20}\) L per cup.

Exercise 10. Baudhāyana bricks: area \( 7\frac12\), brick side \( \frac15\). Number of bricks?

Exact \(= \frac{15}{2}\div \frac{1}{25}=\frac{375}{2}=187.5\). Practically need 188 whole bricks to cover fully.

Exercise 11. Four fountains: 1 day, \( \frac12\) day, \( \frac14\) day, \( \frac15\) day. Time together?

Rates sum to \(12\) cisterns/day ⇒ time \(=\frac{1}{12}\) day = 2 hours.

Exercise 12. Fractional relations (Fig. ideas): If top-right square is \( \frac14\), triangle half of it, shaded \( \frac34\) of triangle. What fraction of whole is shaded?

\( \frac{3}{4}\times \frac{1}{2}\times \frac{1}{4}=\frac{3}{32} \).

Exercise 13. “Dramma-tic donation” value in cowries if 1 dramma = 1280 cowries.

Product \(=\frac{1}{1280}\) of dramma ⇒ 1 cowrie.

Exercise 14. Choose expressions (concept check):
(a) Maria used \( \frac14\) m per bag from 8 m and finished lace. Number of bags?
(b) \( \frac12\) m ribbon makes 8 badges. Ribbon per badge?
(c) Bread: \( \frac16\) kg flour per loaf; 5 kg flour. Number of loaves?

• (a) \( 8\div \frac{1}{4}=32\) bags.
• (b) \( \frac{1}{2}\div 8=\frac{1}{16}\) m per badge.
• (c) \( 5\div \frac{1}{6}=30\) loaves.

Exercise 15. If \( \frac14\) kg flour makes 12 rotis, flour for 6 rotis?

Proportional: \( \frac{1}{4}\times \frac{6}{12}=\frac{1}{8}\) kg.

Exercise 16. Sridharacharya’s sum: \( 1\div \frac16 + 1\div \frac{1}{10} + 1\div \frac{1}{13} + 1\div \frac{1}{9} + 1\div \frac{1}{2}\)

Equals \( 6+10+13+9+2=40 \).

Exercise 17. Novel 400 pages; read \( \frac{1}{5}\) yesterday and \( \frac{3}{10}\) today. Pages left?

• Read \( \frac{1}{5}+\frac{3}{10}=\frac{2}{10}+\frac{3}{10}=\frac{1}{2}\).
• Left \(=\frac{1}{2}\times 400=200\) pages.

Exercise 18. Car 16 km/l. Distance with \( 2\frac{3}{4}\) L?

\( \frac{11}{4}\times 16=44\) km.

Exercise 19. Travel: Train \( 5\frac{1}{6}\) h, Plane \( \frac{1}{2}\) h. Time saved?

\( \frac{31}{6}-\frac{1}{2}=\frac{31}{6}-\frac{3}{6}=\frac{28}{6}=\frac{14}{3}=4\frac{2}{3}\) h.

Exercise 20. Cake: \( \frac{4}{5}\) finished; remaining shared among 3 friends equally. Share per friend?

Remaining \(=\frac{1}{5}\). Each gets \( \frac{1}{5}\div 3=\frac{1}{15}\) of whole cake.

Exercise 21. Choose the correct statements for \( \left(\frac{565}{465}\times \frac{707}{676}\right)\).

• Both factors \(>1\) ⇒ product \(>1\) and \(>\) each factor.

Exercise 22. Evaluate \( (1-\frac12),\ (1-\frac12)(1-\frac13),\dots,(1-\frac12)\cdots(1-\frac{1}{10})\). Make a general statement.

• \(1-\frac12=\frac12\).
• \((1-\frac12)(1-\frac13)=\frac12\cdot \frac23=\frac13\).
• In general, \(\displaystyle\prod_{k=2}^{n}\left(1-\frac{1}{k}\right)=\prod_{k=2}^{n}\frac{k-1}{k}=\frac{1}{n}\).\
• For \(n=10\): value \(=\frac{1}{10}\).