Exercise practice 1: Fractions in Disguise

Fractions in Disguise22 min readLesson 3 of 7

3.1 Figure it Out: Fractions as percentages (pages 3 and 4) The first set of questions asks you to move between fractions and percentages, to compare percentages without calculating, and to reason with a bar model. Read each statement, try it yourself, then compare with the steps below.

1.1 Question 1: Write each fraction as a percentage Statement. Turn each of these fractions into a percentage: 3/5, 7/14, 9/20, 72/150, 1/3 and 5/11. Answer. (i) 60%, (ii) 50%, (iii) 45%, (iv) 48%, (v) 33.33% (33 1/3%), (vi) 45.45% (45 5/11%). Steps. (i) Multiply by 100: 3/5 × 100 = 60. Equivalently 3/5 = 60/100. (ii) Simplify first: 7/14 = 1/2, and 1/2 = 50/100 = 50%. (iii) 9/20 = 9 × 5 / (20 × 5) = 45/100 = 45%. (iv) 72/150 × 100 = 7200/150 = 48. Check: 72/150 = 24/50 = 48/100. (v) 1/3 × 100 = 100/3 = 33.33%. The exact value is 33 1/3%. (vi) 5/11 × 100 = 500/11 = 45.45%. The exact value is 45 5/11%.

1.1 Question 2: White marbles Statement. Nandini has 25 marbles and 15 of them are white. What percentage of her marbles is white? Pick the right option: 10%, 15%, 25%, 60%, 40%, or none of these. Answer. (iv) 60%. Steps. The fraction that is white is 15/25. Simplify: 15/25 = 3/5. Convert to a percentage: 3/5 × 100 = 60%. A common slip is to answer 15%, which is the count of white marbles, not the percentage.

1.1 Question 3: Students who walk to school Statement. A school has 80 students, and 15 of them walk to school each day. What percentage of the students walks? Answer. 18.75%. Steps. Fraction walking = 15/80. Percentage = 15/80 \times 100 = 1500/80. Divide: 1500/80 = 18.75. Check: 15/80 = 3/16, and 3/16 of 100 is 18.75.

1.1 Question 4: The long-distance run Statement. Four friends A, B, C and D are running a race. Their positions after 15 minutes are marked on a track from Start to Finish. Match each runner with the approximate percentage of the race completed: 55%, 20%, 38%, 72%, 84%, 93%. Answer. A is about 38%, B is about 55%, C is about 72%, D is about 93%. Steps. The whole track from Start to Finish represents 100%. A is roughly a third of the way, so the nearest card is 38%. B is just past the halfway mark, so 55%. C is about three quarters of the way, so 72%. D is close to the finish line, so 93%. The cards 20% and 84% do not match any runner: no one has covered only one fifth of the track, and no one is at 84%.

1.1 Question 5: Put >, < or = Statement. Compare these pairs and fill in >, < or = in place of the blanks: 50% against 5%; 5/10 against 50%; 3/11 against 61%; 30% against 1/3. Decide without calculating where you can. Answer. (i) >, (ii) =, (iii) <, (iv) <. Steps. (i) Both percentages refer to the same whole, and 50 is greater than 5, so 50% > 5%. (ii) 5/10 simplifies to 1/2, and 50% is also 1/2, so they are equal. (iii) 3/11 is a little more than 3/12 = 1/4 = 25%, and 61% is more than twice that, so 3/11 < 61%. (iv) 1/3 = 33.33%, which is greater than 30%, so 30% < 1/3.

3.2 Figure it Out: Percentage of a quantity (pages 12 to 14) This set mixes bar models, mental computation and proportional reasoning. The estimate comes first in every question; the exact calculation confirms it.

1.2 Question 1: Find the missing numbers in the bar models Statement. Each bar is divided into equal cells and stands for 100%. Find the missing numbers. (i) A bar of 5 cells has 20% marked on one cell. A second bar of 5 cells shows 75 at its end, and 60 is marked against four cells. (ii) A bar of 10 cells has one cell marked with an unknown percentage. A second bar of 10 cells shows 90 at its end, and an unknown value is marked against six cells. (iii) A bar of 4 cells has one cell marked with an unknown percentage. A second bar of 4 cells shows 140 at its end, and an unknown value is marked against three cells. Answer. (i) one cell is 20% and the value 20% of 75 is 15; 60 is 80% of 75. (ii) one cell is 10%, and six cells are 60%, which is 54. (iii) one cell is 25%, its value is 35, and three cells are 75%, which is 105. Steps. Divide 100% by the number of cells to get the share of one cell, then multiply by the value of the whole bar. (i) 100%/5 = 20% per cell. The whole bar is 75, so one cell is 20% of 75 = 15. The 60 bracket covers 4 cells: 4 \times 20% = 80%, and 80% of 75 = 60, which checks. (ii) 100%/10 = 10% per cell. The whole bar is 90, so one cell is 10% of 90 = 9. The bracket covers 6 cells: 60% of 90 = 54. (iii) 100%/4 = 25% per cell. The whole bar is 140, so one cell is 25% of 140 = 35. The bracket covers 3 cells: 75% of 140 = 105. A bar model turns a percentage into a picture: the whole bar is 100%, and equal cells split it into convenient pieces.

1.2 Question 2: Value plus bar model Statement. Work out 25% of 160, 16% of 250, 62% of 360, 140% of 40, 1% of 1 hour and 7% of 10 kg. Draw a bar model for each part. Answer. (i) 40, (ii) 40, (iii) 223.2, (iv) 56, (v) 36 seconds, (vi) 0.7 kg. Steps. (i) 25% = 1/4, so 160/4 = 40. Bar model: four equal cells, one cell shaded. (ii) 16% = 10% + 5% + 1%: 25 + 12.5 + 2.5 = 40. (iii) 62% = 60% + 2%: 0.6 \times 360 = 216, 0.02 \times 360 = 7.2, total 223.2. Bar model: 100 cells, 62 shaded; the value sits a little above 60% of the bar. (iv) 140% = 100% + 40%: 40 + 0.4 \times 40 = 40 + 16 = 56. The bar model is one full bar plus 40% of a second, that is 1.4 bars. (v) 1 hour = 60 minutes, so 1% of 60 = 0.6 minutes, which is 36 seconds. (vi) 1% of 10 kg = 0.1 kg, so 7% = 0.7 kg.

1.2 Question 3: Red paint in the mixture Statement. Surya mixed 60 ml of deep orange paint, with red making up three quarters of the mixture. What volume of red paint went in? Answer. 45 ml. Steps. The mixture is 60 ml and red is 3/4 of it. 60 \times 3/4 = 180/4 = 45. Check: 45/60 = 3/4 = 75%, so the red share is 75% and the yellow share is 15 ml.

1.2 Question 4: Compare the pairs Statement. Compare each pair and fill in >, < or = where the blank appears: 50% of 510 and 50% of 515; 37% of 148 and 73% of 148; 29% of 43 and 92% of 110; 30% of 40 and 40% of 50; 45% of 200 and 10% of 490; 30% of 80 and 24% of 64. Estimate first, compute only to confirm. Answer. (i) <, (ii) <, (iii) <, (iv) <, (v) >, (vi) >. Steps. (i) Same percentage, smaller whole: 50% of 510 < 50% of 515. (ii) Same whole, smaller percentage: 37% of 148 < 73% of 148. (iii) 29% of 43 is about 30% of 43, about 13. 92% of 110 is about 100, clearly bigger. Computed: 12.47 < 101.2. (iv) 30% of 40 = 12 and 40% of 50 = 20, so <. (v) 45% of 200 = 90 and 10% of 490 = 49, so >. (vi) 30% of 80 = 24 and 24% of 64 = 15.36, so >.

1.2 Question 5: Fill in the blanks Statement. (i) 30% of k is 70; find 60% of k, 90% of k and 120% of k. (ii) 100% of m is 215; find 10% of m, 1% of m and 6% of m. (iii) 90% of n is 270; find 9% of n, 18% of n and 100% of n. (iv) Make two more such questions and challenge a friend. Answer. (i) 140, 210, 280. (ii) 21.5, 2.15, 12.9. (iii) 27, 54, 300. Steps. (i) 30% of k is 70, so 10% of k is 70/3, and k itself is 700/3. Then 60% = 6 \times 70/3 = 140, 90% = 9 \times 70/3 = 210, and 120% = 12 \times 70/3 = 280. (ii) Start with 100% = 215. Then 10% = 21.5, 1% = 2.15, and 6% = 6 \times 2.15 = 12.9. (iii) 90% of n is 270, so 10% of n is 30 and n = 300. Then 9% = 9 \times 3 = 27, 18% = 54, and 100% = 300. (iv) Two sample questions with answers: "40% of t is 120. Find 20% of t and 120% of t." The answers are 60 and 360. "75% of u is 45. Find 25% of u and 100% of u." The answers are 15 and 60.

1.2 Question 6: Fill in the blanks Statement. Fill each blank: 3 is what percentage of 300? Which number is 40% of 4? 40 is 80% of which number? Answer. (i) 1%, (ii) 1.6, (iii) 50. Steps. (i) 3/300 \times 100 = 1. So 3 is 1% of 300. (ii) 40% of 4 = 0.4 \times 4 = 1.6. (iii) Let the unknown be x. Then 0.8x = 40, so x = 40/0.8 = 50.

1.2 Question 7: A day and a week Statement. Which is longer, 10% of one day or 1% of one week? Make up similar questions and ask your classmates. Answer. 10% of a day is longer: it is 2.4 hours, while 1% of a week is 1.68 hours. Steps. 10% of a day = 0.1 \times 24 = 2.4 hours = 2 hours 24 minutes. 1% of a week = 0.01 \times 7 days = 0.07 days = 0.07 \times 24 = 1.68 hours = about 1 hour 41 minutes. 2.4 > 1.68, so the first quantity is longer. Similar questions: "Is 50% of an hour longer than 5% of a day?" The answers are 30 minutes and 72 minutes, so no. "Is 20% of a week longer than 50% of a day?" The answers are 33.6 hours and 12 hours, so yes.

1.2 Question 8: The peculiar bull Statement. Mariam's bull is given 2 units of fodder on the first day and eats 1 unit. On day 2 it is given 3 units and eats 2. On day 3 it is given 4 units and eats 3. This continues, and on day 99 it is given 100 units and eats 99. Write the eaten quantity as a percentage of the given quantity and describe what you observe. Answer. The percentages are 50%, 66.67%, 75%, and so on up to 99%. They increase towards 100% but never reach it. Steps. On day n the bull is given n + 1 units and eats n units, so the percentage eaten is n/(n + 1) \times 100. Day 1: 1/2 = 50%. Day 2: 2/3 = 66.67%. Day 3: 3/4 = 75%. Day 4: 4/5 = 80%. Day 9: 9/10 = 90%. Day 49: 49/50 = 98%. Day 99: 99/100 = 99%. The percentage rises each day, because the extra unit eaten is a larger share of a larger total. It always stays below 100%, since one unit is never eaten. The gains become smaller as the days go on.

1.2 Question 9: Picking coffee berries Statement. Workers take 18 days to pick berries on 20% of a plantation. How long will the whole plantation take if the pace of work does not change? Why is that assumption important? Answer. 90 days. Steps. 20% of the plantation takes 18 days. 1% takes 18/20 = 0.9 days. 100% takes 0.9 \times 100 = 90 days. The assumption matters because the answer is a proportion only if each day's work covers the same share. If the workers slow down, take breaks, or the ripe berries are spread unevenly, the time would change.

1.2 Question 10: The badminton training plan Statement. A coach plans training with warm up : play : cool down in the ratio 10% : 80% : 10%. How long should each part of a 90-minute session last? Answer. Warm up 9 minutes, play 72 minutes, cool down 9 minutes. Steps. The percentages add to 10 + 80 + 10 = 100, so they describe the whole session. Warm up = 10% of 90 = 9 minutes. Play = 80% of 90 = 72 minutes. Cool down = 10% of 90 = 9 minutes. Check: 9 + 72 + 9 = 90.

1.2 Question 11: People in the Northern Hemisphere Statement. About 90% of the world's people live in the Northern Hemisphere. Use this year's world population to estimate how many people that is. Answer. About 7.38 billion, or roughly 7.4 billion. Steps. The chapter uses a 2025 world population of about 8.2 billion. 90% of 8.2 billion = 0.9 \times 8.2 = 7.38 billion. Rounded to one decimal place, about 7.4 billion people.

1.2 Question 12: Halwa for four and for eight Statement. A halwa recipe for 4 people has rava 40%, sugar 40% and ghee 20%. For 8 people, what is each ingredient's proportion? If all the ingredients together weigh 2 kg, find the weight of rava, sugar and ghee. Answer. (i) The proportions do not change: rava 40%, sugar 40%, ghee 20%. (ii) Rava 800 g, sugar 800 g, ghee 400 g. Steps. (i) Doubling the number of people doubles every quantity, so each ingredient stays the same share of the mixture. Percentages describe shares, not amounts. (ii) 2 kg = 2000 g. Rava = 40% of 2000 = 800 g. Sugar = 40% of 2000 = 800 g. Ghee = 20% of 2000 = 400 g. Check: 800 + 800 + 400 = 2000 g, and the percentages add to 100%.