Skip to main content

Grade 10 Mathematics Teachers’ Book

INNODEMS

Subsection 1.1.2 Reciprocals of Numbers

In this subsection, we explore reciprocals of real numbers using division, mathematical tables and calculators, and apply reciprocals in mathematical computations.
🔗

Subsubsection 1.1.2.1 Reciprocals using Division

Learner Experience 1.1.4.

Work in groups: Form groups of \(2\) or \(3\) students
🔗
Materials: Paper, a ruler, and scissors.
🔗
Instructions:
🔗
  1. We’re going to cut out different rectangles that have the same area.
    1. Cut out a square of length 1cm. (If your ruler has inches on it, it might be easier to cut out a square of length 1 inch.)
      🔗
      🔗
    2. Cut out a rectangle with length 2cm and height 0.5cm.
      🔗
      🔗
    3. Cut out a rectangle with length 3cm and height 0.33cm.
      🔗
      🔗
    4. Cut out a rectangle with height 0.5cm and length 2cm.
      🔗
      🔗
    🔗
    🔗
  2. Arrange your rectangles so that their bottom left corners are on top of each other, and their lengths are parralel, like the black boxes in Figure 1.1.23
    🔗
    🔗
  3. Visually confirm that all your rectangles have roughly the same area as the square (which is \(1 \text{ cm}^2\)).
    🔗
    🔗
  4. Consider the following interactive:
    🔗
    Figure 1.1.23. Using \(y = \frac{1}{x}\) to find reciprocals
    🔗
    As you drag the point \(P\text{,}\) the red rectangle has length \(x\) and height \(\frac{1}{x}\text{.}\) Given that the area remains constant at \(1 \text{ cm}^2\text{,}\) and (as we will cover in Section 2.5), area is length \(\times\) height, what do you know about \(x \times \frac{1}{x}\text{?}\)
    🔗
    🔗
  5. If a rectangle has length 2.5cm, but area \(1 \text{ cm}^2\) (like the other rectangles), what must its height be? Use the figure to find out.
    🔗
    🔗
  6. Write 2.5 as a fraction, as well as your previous answer. What do you notice about these fractions? Does this generalise to other fractions?
    🔗
    🔗
🔗
🔗

Key Takeaway 1.1.24.

The reciprocal of a number is the number that, when multiplied by the original number, gives a product of \(1\text{.}\)
🔗
In other words the reciprocal of a real number \(x\) is \(\frac{1}{x}\text{,}\) except when \(x=0\) since division by zero is undefined.
🔗
Taking the reciprocal of a fraction swaps the numerator and denominator: \(1 \div \frac{a}{b} = \frac{1}{\frac{a}{b}} = \frac{b}{a}\)
🔗
For example:
  1. The reciprocal of \(2\) is \(\frac{1}{2}\)
    🔗
    🔗
  2. The reciprocal of \(\frac{3}{5}\text{,}\) using division, is \(1 \div \frac{3}{5} = \frac{5}{3}\text{.}\) You can find this using long division.
    🔗
    🔗
🔗
To find the reciprocal of a real number, we can use long division
🔗
🔗

Example 1.1.25.

Find the reciprocal of \(5\)
🔗
Solution.
Using long division:
\(0.2\)
\(5\) \(10\)
\(-\) \(10\)
\(0\)
🔗
Which is \(= 0.2\)
🔗
🔗
🔗

Example 1.1.26.

Find the reciprocal (to 5 decimal places) of the following numbers using long division.
🔗
  1. \(\displaystyle 11\)
    🔗
    🔗
  2. \(\displaystyle 4.2\)
    🔗
    🔗
🔗
Solution.
  1. The reciprocal of \(11\) is \(\frac{1}{11}\text{,}\) so we need to do long devision of 1 divided by 11.
    🔗
    Hence, the reciprocal of \(11\) using division is \(1\div 11 = 0.090909\ldots\text{,}\) which is \(0.09091\) to five decimal places.
    🔗
    🔗
  2. The reciprocal of \(4.2\) according to definition is \(\frac{1}{4.2}\text{.}\)
    🔗
    We can rerwrite \(\frac{1}{4.2}\) as \(\frac{1}{4.2}\times \frac{10}{10} \)
    \begin{align*} =\amp \frac{10}{42} \\ =\amp \frac{5}{21} \end{align*}
    🔗
    Dividing \(5\) by \(21\) we have:
    🔗
    Therefore, the reciprocal of \(4.2\) given to 4 decimal places is \(0.2381\)
    🔗
    🔗
🔗
🔗
🔗

Example 1.1.27.

If \(\frac{1}{x} = 0.7\text{,}\) find \(x - \frac{1}{x}\) to 3 decimal places.
🔗
Solution.
Given \(\frac{1}{x} = 0.7\text{,}\) we compute \(x\) as the reciprocal of \(0.7\) using division: \(x = 1 \div 0.7 = 10 \div 7\text{.}\)
🔗
Hence \(x = 1.4285\ldots\text{,}\) which to three decimal places is \(1.429\text{.}\)
🔗
Now compute the required expression: \(x - \frac{1}{x} = 1.429 - 0.7 = 0.729\text{.}\)
🔗
Therefore \(x - \frac{1}{x} = 0.729\) (to three decimal places).
🔗
🔗
🔗

Checkpoint 1.1.28.

🔗

Checkpoint 1.1.29.

🔗

Checkpoint 1.1.30.

🔗

Exercises Exercise

1.
Using division, find the reciprocal of following numbers:
🔗
  1. \(\displaystyle 5\)
    🔗
    🔗
  2. \(\displaystyle -3\)
    🔗
    🔗
  3. \(\displaystyle 2.5\)
    🔗
    🔗
  4. \(\displaystyle -0.4\)
    🔗
    🔗
  5. \(\displaystyle \frac{2}{7}\)
    🔗
    🔗
  6. \(\displaystyle -\frac{5}{8}\)
    🔗
    🔗
🔗
Answer.
  1. \(\displaystyle \frac{1}{5} = 0.2\)
    🔗
    🔗
  2. \(\displaystyle \frac{1}{-3} = -0.333...\)
    🔗
    🔗
  3. \(\displaystyle \frac{1}{2.5} = 0.4\)
    🔗
    🔗
  4. \(\displaystyle \frac{1}{-0.4} = -2.5\)
    🔗
    🔗
  5. \(\displaystyle \frac{1}{\frac{2}{7}} = 3.5\)
    🔗
    🔗
  6. \(\displaystyle \frac{1}{-\frac{5}{8}} = -1.6\)
    🔗
    🔗
🔗
🔗
🔗
2.
If the reciprocal of \(x\) is \(\frac{1}{6}\text{,}\) find the value of \(x\text{.}\)
🔗
Answer.
\(\frac{1}{x} = \frac{1}{6}\text{.}\) Therefore, \(x = 6\text{.}\)
🔗
🔗
🔗
3.
Movin, a Grade \(10\) learner covers \(\frac{1}{x}\) km from home to school. If the distance he covers each day from home to school is \(0.2\) km. Find the value of \(x\text{.}\)
🔗
Answer.
\(\frac{1}{x} = \frac{2}{10} = \frac{1}{5}\) which implies that \(x = 5\text{.}\)
🔗
🔗
🔗
4.
A cyclist covers a distance of \(12 \) km in \(1\) hour. Using division to find reciprocals, determine the time taken to cover \(1\) km at the same speed.
🔗
Answer.
The reciprocal of \(12\) is \(1 \div 12 = 0.0833\) hours.
🔗
🔗
🔗
5.
A factory machine produces \(\frac{5}{8}\) of a widget every minute. How long will it take the machine to produce \(40\) widgets?
🔗
Answer.
The reciprocal of \(\frac{5}{8}\) is \(1 \div \frac{5}{8} = 1.6\) minutes per widget. Time taken to produce \(40\) widgets is \(64\) minutes.
🔗
🔗
🔗
6.
A typist can type \(\frac{4}{7}\) of a page in \(8\) minutes. At this rate, how long will it take to type a full \(25\)-page document?
🔗
Answer.
The reciprocal of \(\frac{4}{7}\) is \(1 \div \frac{4}{7} = 1.75\) minutes per page. Time taken to type \(25\) pages is \(43.75\) minutes.
🔗
🔗
🔗
🔗
🔗

Subsubsection 1.1.2.2 Reciprocals Using Tables and Calculators

Mathematical tables and calculators provide alternative methods for quickly determining reciprocals, especially when dealing with large numbers or decimal values. In this section, we will explore how to use both tools effectively.
🔗

Finding reciprocals of numbers using mathematical tables.

🔗
Reciprocal tables are specially designed tables that allow you to quickly find the reciprocal of a number without doing any division. Each table lists a number along with its reciprocal, so you can look up the reciprocal directly.
🔗
Table 1.1.32. Reciprocal Table
🔗
SUBTRACT
\(x\) 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9
1.0 1.0000 9901 9804 9709 9615 9524 9434 9346 9259 9174 9 18 28 37 46 55 64 73 83
1.1 0.9091 9009 8929 8850 8772 8696 8621 8547 8475 8403 8 15 23 31 38 46 54 61 69
1.2 0.8333 8264 8197 8130 8065 8000 7937 7874 7813 7752 6 13 19 26 32 39 45 52 58
1.3 0.7692 7634 7576 7519 7463 7407 7353 7299 7246 7194 6 11 17 22 28 33 39 44 50
1.4 0.7143 7092 7042 6993 6944 6897 6849 6803 6757 6711 5 10 14 19 24 29 34 38 43
1.5 0.6667 6623 6579 6536 6494 6452 6410 6369 6329 6289 4 8 13 17 21 25 29 34 38
1.6 0.6250 6211 6173 6135 6098 6061 6024 5988 5952 5917 4 7 11 15 19 22 26 30 33
1.7 0.5882 5848 5814 5780 5747 5714 5682 5650 5618 5587 3 7 10 13 16 20 23 26 30
1.8 0.5556 5525 5495 5464 5435 5405 5376 5348 5319 5291 3 6 9 12 15 18 21 24 27
1.9 0.5263 5236 5208 5181 5155 5128 5102 5076 5051 5025 3 5 8 11 13 16 19 21 24
2.0 0.5000 4975 4950 4926 4902 4878 4854 4831 4808 4785 2 5 7 10 12 14 17 19 22
2.1 0.4762 4739 4717 4695 4673 4651 4630 4608 4587 4566 2 4 7 9 11 13 15 17 20
2.2 0.4545 4525 4505 4484 4464 4444 4425 4405 4386 4367 2 4 6 8 10 12 14 16 18
2.3 0.4348 4329 4310 4292 4274 4255 4237 4219 4202 4184 2 4 5 7 9 11 13 15 16
2.4 0.4167 4149 4132 4115 4098 4082 4065 4049 4032 4016 2 3 5 7 8 10 12 13 15
2.5 0.4000 3984 3968 3953 3937 3922 3906 3891 3876 3861 2 3 5 6 8 9 11 12 14
2.6 0.3846 3831 3817 3802 3788 3774 3759 3745 3731 3717 1 3 4 6 7 9 10 11 13
2.7 0.3704 3690 3676 3663 3650 3636 3623 3610 3597 3584 1 3 4 5 7 8 9 11 12
2.8 0.3571 3559 3546 3534 3521 3509 3497 3484 3472 3460 1 2 4 5 6 7 9 10 11
2.9 0.3448 3436 3425 3413 3401 3390 3378 3367 3356 3344 1 2 3 5 6 7 8 9 10
3.0 0.3333 3322 3311 3300 3289 3279 3268 3257 3247 3236 1 2 3 4 5 6 8 9 10
3.1 0.3226 3215 3205 3195 3185 3175 3165 3155 3145 3135 1 2 3 4 5 6 7 8 9
3.2 0.3125 3115 3106 3096 3086 3077 3067 3058 3049 3040 1 2 3 4 5 6 7 8 9
3.3 0.3030 3021 3012 3003 2994 2985 2976 2967 2959 2950 1 2 3 4 4 5 6 7 8
3.4 0.2941 2933 2924 2915 2907 2899 2890 2882 2874 2865 1 2 3 3 4 5 6 7 8
3.5 0.2857 2849 2841 2833 2825 2817 2809 2801 2793 2786 1 2 2 3 4 5 6 6 7
3.6 0.2778 2770 2762 2755 2747 2740 2732 2725 2717 2710 1 2 2 3 4 5 5 6 7
3.7 0.2703 2695 2688 2681 2674 2667 2660 2653 2646 2639 1 1 2 3 4 4 5 6 6
3.8 0.2632 2625 2618 2611 2604 2597 2591 2584 2577 2571 1 1 2 3 3 4 5 5 6
3.9 0.2564 2558 2551 2545 2538 2532 2525 2519 2513 2506 1 1 2 3 3 4 5 5 6
4.0 0.2500 2494 2488 2481 2475 2469 2463 2457 2451 2445 1 1 2 2 3 4 4 5 6
4.1 0.2439 2433 2427 2421 2415 2410 2404 2398 2392 2387 1 1 2 2 3 3 4 5 5
4.2 0.2381 2375 2370 2364 2358 2353 2347 2342 2336 2331 1 1 2 2 3 3 4 4 5
4.3 0.2326 2320 2315 2309 2304 2299 2294 2288 2283 2278 1 1 2 2 3 3 4 4 5
4.4 0.2273 2268 2262 2257 2252 2247 2242 2237 2232 2227 1 1 2 2 3 3 4 4 5
4.5 0.2222 2217 2212 2208 2203 2198 2193 2188 2183 2179 0 1 1 2 2 3 3 4 4
4.6 0.2174 2169 2165 2160 2155 2151 2146 2141 2137 2132 0 1 1 2 2 3 3 4 4
4.7 0.2128 2123 2119 2114 2110 2105 2101 2096 2092 2088 0 1 1 2 2 3 3 4 4
4.8 0.2083 2079 2075 2070 2066 2062 2058 2053 2049 2045 0 1 1 2 2 3 3 3 4
4.9 0.2041 2037 2033 2028 2024 2020 2016 2012 2008 2004 0 1 1 2 2 2 3 3 4
5.0 0.2000 1996 1992 1988 1984 1980 1976 1972 1969 1965 0 1 1 2 2 2 3 3 4
5.1 0.1961 1957 1953 1949 1946 1942 1938 1934 1931 1927 0 1 1 2 2 2 3 3 3
5.2 0.1923 1919 1916 1912 1908 1905 1901 1898 1894 1890 0 1 1 1 2 2 3 3 3
5.3 0.1887 1883 1880 1876 1873 1869 1866 1862 1859 1855 0 1 1 1 2 2 2 3 3
5.4 0.1852 1848 1845 1842 1838 1835 1832 1828 1825 1821 0 1 1 1 2 2 2 3 3
5.5 0.1818 1815 1812 1808 1805 1802 1799 1795 1792 1789 0 1 1 1 2 2 2 3 3
5.6 0.1786 1783 1779 1776 1773 1770 1767 1764 1761 1757 0 1 1 1 2 2 2 3 3
5.7 0.1754 1751 1748 1745 1742 1739 1736 1733 1730 1727 0 1 1 1 2 2 2 2 3
5.8 0.1724 1721 1718 1715 1712 1709 1706 1704 1701 1698 0 1 1 1 1 2 2 2 3
5.9 0.1695 1692 1689 1686 1684 1681 1678 1675 1672 1669 0 1 1 1 1 2 2 2 3
6.0 0.1667 1664 1661 1658 1656 1653 1650 1647 1645 1642 0 1 1 1 1 2 2 2 3
6.1 0.1639 1637 1634 1631 1629 1626 1623 1621 1618 1616 0 1 1 1 1 2 2 2 2
6.2 0.1613 1610 1608 1605 1603 1600 1597 1595 1592 1590 0 1 1 1 1 2 2 2 2
6.3 0.1587 1585 1582 1580 1577 1575 1572 1570 1567 1565 0 0 1 1 1 1 2 2 2
6.4 0.1563 1560 1558 1555 1553 1550 1548 1546 1543 1541 0 0 1 1 1 1 2 2 2
6.5 0.1538 1536 1534 1531 1529 1527 1524 1522 1520 1517 0 0 1 1 1 1 2 2 2
6.6 0.1515 1513 1511 1508 1506 1504 1502 1499 1497 1495 0 0 1 1 1 1 2 2 2
6.7 0.1493 1490 1488 1486 1484 1481 1479 1477 1475 1473 0 0 1 1 1 1 2 2 2
6.8 0.1471 1468 1466 1464 1462 1460 1458 1456 1453 1451 0 0 1 1 1 1 2 2 2
6.9 0.1449 1447 1445 1443 1441 1439 1437 1435 1433 1431 0 0 1 1 1 1 1 2 2
7.0 0.1429 1427 1425 1422 1420 1418 1416 1414 1412 1410 0 0 1 1 1 1 1 2 2
7.1 0.1408 1406 1404 1403 1401 1399 1397 1395 1393 1391 0 0 1 1 1 1 1 2 2
7.2 0.1389 1387 1385 1383 1381 1379 1377 1376 1374 1372 0 0 1 1 1 1 1 2 2
7.3 0.1370 1368 1366 1364 1362 1361 1359 1357 1355 1353 0 0 1 1 1 1 1 2 2
7.4 0.1351 1350 1348 1346 1344 1342 1340 1339 1337 1335 0 0 1 1 1 1 1 1 2
7.5 0.1333 1332 1330 1328 1326 1325 1323 1321 1319 1318 0 0 1 1 1 1 1 1 2
7.6 0.1316 1314 1312 1311 1309 1307 1305 1304 1302 1300 0 0 1 1 1 1 1 1 2
7.7 0.1299 1297 1295 1294 1292 1290 1289 1287 1285 1284 0 0 1 1 1 1 1 1 2
7.8 0.1282 1280 1279 1277 1276 1274 1272 1271 1269 1267 0 0 1 1 1 1 1 1 2
7.9 0.1266 1264 1263 1261 1259 1258 1256 1255 1253 1252 0 0 0 1 1 1 1 1 1
8.0 0.1250 1248 1247 1245 1244 1242 1241 1239 1238 1236 0 0 0 1 1 1 1 1 1
8.1 0.1235 1233 1232 1230 1229 1227 1225 1224 1222 1221 0 0 0 1 1 1 1 1 1
8.2 0.1220 1218 1217 1215 1214 1212 1211 1209 1208 1206 0 0 0 1 1 1 1 1 1
8.3 0.1205 1203 1202 1200 1199 1198 1196 1195 1193 1192 0 0 0 1 1 1 1 1 1
8.4 0.1190 1189 1188 1186 1185 1183 1182 1181 1179 1178 0 0 0 1 1 1 1 1 1
8.5 0.1176 1175 1174 1172 1171 1170 1168 1167 1166 1164 0 0 0 1 1 1 1 1 1
8.6 0.1163 1161 1160 1159 1157 1156 1155 1153 1152 1151 0 0 0 1 1 1 1 1 1
8.7 0.1149 1148 1147 1145 1144 1143 1142 1140 1139 1138 0 0 0 0 1 1 1 1 1
8.8 0.1136 1135 1134 1133 1131 1130 1129 1127 1126 1125 0 0 0 0 1 1 1 1 1
8.9 0.1124 1122 1121 1120 1119 1117 1116 1115 1114 1112 0 0 0 1 1 1 1 1 1
9.0 0.1111 1110 1109 1107 1106 1105 1104 1103 1101 1100 0 0 0 0 1 1 1 1 1
9.1 0.1099 1098 1096 1095 1094 1093 1092 1091 1089 1088 0 0 0 0 1 1 1 1 1
9.2 0.1087 1086 1085 1083 1082 1081 1080 1079 1078 1076 0 0 0 0 1 1 1 1 1
9.3 0.1075 1074 1073 1072 1071 1070 1068 1067 1066 1065 0 0 0 0 1 1 1 1 1
9.4 0.1064 1063 1062 1060 1059 1058 1057 1056 1055 1054 0 0 0 0 1 1 1 1 1
9.5 0.1053 1052 1050 1049 1048 1047 1046 1045 1044 1043 0 0 0 0 1 1 1 1 1
9.6 0.1042 1041 1040 1038 1037 1036 1035 1034 1033 1032 0 0 0 0 1 1 1 1 1
9.7 0.1031 1030 1029 1028 1027 1026 1025 1024 1022 1021 0 0 0 0 1 1 1 1 1
9.8 0.1020 1019 1018 1017 1016 1015 1014 1013 1012 1011 0 0 0 0 1 1 1 1 1
9.9 0.1010 1009 1008 1007 1006 1005 1004 1003 1002 1001 0 0 0 0 1 1 1 1 1

Learner Experience 1.1.5.

Each group should have a mathematical table
🔗
1. Working in groups of \(5\text{,}\) consider the following numbers:
🔗
  1. \(\displaystyle \frac{3}{4}\)
    🔗
    🔗
  2. \(\displaystyle \frac{1}{3}\)
    🔗
    🔗
  3. \(\displaystyle 6\)
    🔗
    🔗
  4. \(\displaystyle 0.4167\)
    🔗
    🔗
🔗
2. Discuss how to use reciprocal tables to find reciprocals of each of the given numbers.
🔗
3. Determine the reciprocals of the numbers.
🔗
4. What did you realize?
🔗
5. Share your work with fellow learners.
🔗
🔗

Key Takeaway 1.1.33.

How to find reciprocals of numbers from the table.
🔗
1. Given a large number e.g \(1252\text{,}\) express the number in standard form: \(1.252 \times 10^3\text{.}\)
🔗
2. Next, you find the reciprocal of the number from the reciprocal table as follows.
🔗
  1. Move down the column headed \(x\) to locate \(1.2\)
    🔗
    🔗
  2. Move to the right along the row that has \(1.2\) to the column headed \(5\text{.}\)
    🔗
    🔗
  3. Read the number at the intersection of the row and column which is \(0.8000\)
    🔗
    🔗
  4. Move farther to the right on the same row to the SUBTRACT column headed \(2\text{.}\)
    🔗
    🔗
  5. Read the number at the intersection of the row and column which is \(13\text{.}\)
    🔗
    🔗
  6. Subtract the number \(13\) from \(0.800\text{.}\) While subtracting align \(13\) to the right as shown below.
    🔗
    \(0\) \(.\) \(8\) \(0\) \(0\)
    - \(0\) \(.\) \(0\) \(1\) \(3\)
    \(0\) \(.\) \(7\) \(8\) \(7\)
    🔗
🔗
4. Calculate the reciprocal of \(10^3\text{.}\)
🔗
Since \(10^3\) can be written as \(1000\) then its reciprocal is \(\frac{1}{1000}\text{.}\)
🔗
5. Multiply \(0.787\) by \(\frac{1}{1000}\text{.}\)
🔗
6. Hence, the reciprocal of \(1252\) is given by \(0.787 \times \frac{1}{1000} = 0.000787\)
🔗
🔗

Example 1.1.34.

Use tables to find the reciprocal of \(0.154\text{.}\)
🔗
Solution.
Given \(0.154\text{,}\) first write it in standard form: \(1.54 \times 10^{-1}\text{.}\)
🔗
Then, to locate the reciprocal of the given number:
🔗
  1. Move down the column headed \(x\) to locate \(1.5\text{.}\)
    🔗
    🔗
  2. Move to the right along the row that has \(1.5\) to the column headed \(4\text{.}\)
    🔗
    🔗
  3. Read the number at the intersection of the row and column which is \(0.6494\text{.}\)
    🔗
    🔗
🔗
Calculate the reciprocal of \(10^{-1}\text{.}\)
🔗
Since \(10^{-1}\) can be written as \(\frac{1}{10}\) then its reciprocal is \(\frac{10}{1} = 10\text{.}\)
🔗
Multiply \(0.6494\) by \(10\text{.}\)
🔗
The reciprocal of \(0.154\) is \(6.494\)
🔗
🔗
🔗

Example 1.1.35.

Murunga’s car consumes \(\frac{1}{8}\) liters of fuel per kilometer. Use tables to identify how far Murunga can drive with 1 liter?
🔗
Solution.
If \(\frac{1}{8} \text{liters} =1\)km,
🔗
Then \(1 \text{liter}= ?\) km
🔗
To find the distance Murunga drove using \(1\) liter, we first convert \(\frac{1}{8}\) to a decimal i.e \(1\div8 = 0.125\text{.}\)
🔗
Next, we write \(0.125\) in standard form: \(1.25 \times 10^{-1}\text{.}\)
🔗
Move down the column headed \(x\) to locate \(1.2\text{.}\)
🔗
Move to the right along the row that has \(1.2\) to the column headed \(5\text{.}\)
🔗
Read the number at the intersection of the row and column which is \(0.8000\text{.}\)
🔗
Calculate the reciprocal of \(10^{-1}\text{.}\)
🔗
Since \(10^{-1}\) can be written as \(\frac{1}{10}\) then its reciprocal is \(\frac{10}{1} = 10\text{.}\)
🔗
Multiply \(0.8000\) by \(10\text{.}\)
🔗
\(0.8000 \times 10 = 8\)
🔗
Therefore, Murunga can drive \(8\) km using \(1\) l of fuel.
🔗
🔗
🔗

Checkpoint 1.1.36.

🔗

Checkpoint 1.1.37.

🔗

Exercises Exercise

1.
Find the reciprocals of the following numbers using reciprocal tables:
🔗
  1. \(\displaystyle 4286\)
    🔗
    🔗
  2. \(\displaystyle 0.0458\)
    🔗
    🔗
  3. \(\displaystyle 0.007582\)
    🔗
    🔗
  4. \(\displaystyle 2.781\)
    🔗
    🔗
  5. \(\displaystyle \frac{3}{8}\)
    🔗
    🔗
  6. \(\displaystyle \frac{4}{0.125}\)
    🔗
    🔗
🔗
Answer.
  1. \(\displaystyle 0.0002332\)
    🔗
    🔗
  2. \(\displaystyle 21.83\)
    🔗
    🔗
  3. \(\displaystyle 131.9\)
    🔗
    🔗
  4. \(\displaystyle 0.3596\)
    🔗
    🔗
  5. \(\displaystyle 26.67\)
    🔗
    🔗
  6. \(\displaystyle 0.03125\)
    🔗
    🔗
🔗
🔗
🔗
2.
Use mathematical tables to evaluate each of the following:
🔗
  1. \(\displaystyle \frac{100}{29.56}\)
    🔗
    🔗
  2. \(\displaystyle \frac{1}{\sqrt{0.2704}}\)
    🔗
    🔗
  3. \(\displaystyle \frac{1}{1.374^2}\)
    🔗
    🔗
  4. \(\displaystyle 1000 \times \frac{1}{0.7598}\)
    🔗
    🔗
  5. \(\displaystyle \frac{3}{\sqrt{2025}}\)
    🔗
    🔗
  6. \(\displaystyle 3.054 + \frac{1}{60.84}\)
    🔗
    🔗
🔗
Answer.
  1. \(\frac{100}{29.56} = 100 \times \frac{1}{29.56}\)
    🔗
    \(\frac{1}{29.56} = 0.03383\)
    🔗
    \(100 \times 0.03383 =3.383\)
    🔗
    🔗
  2. \(\frac{1}{\sqrt{0.2704}} = \frac{1}{0.52}\)
    🔗
    \(\frac{1}{0.52} = 1.923\)
    🔗
    🔗
  3. \(\frac{1}{1.374^2} = \frac{1}{1.887876} \)
    🔗
    \(\frac{1}{1.888} = 0.5296\)
    🔗
    🔗
  4. \(1000 \times \frac{1}{0.7598} = 1000 \times \frac{1}{7.598 \times 10^{-1}}\)
    🔗
    \(\frac{1}{0.7598} = 1.317\)
    🔗
    \(1000 \times 1.317 = 1317\)
    🔗
    🔗
  5. \(\frac{5}{\sqrt{2025}} = \frac{1}{9}\)
    🔗
    \(= 0.1111\)
    🔗
    🔗
  6. \(3.054 + \frac{1}{60.84} = 3.054 + \frac{1}{6.084 \times 10^{1}}\)
    🔗
    \(\frac{1}{60.84} = 0.01644\)
    🔗
    \(3.054 + 0.01644 = 3.07044\)
    🔗
    🔗
🔗
🔗
🔗
3.
Maria conducted a survey to find out which games students enjoy. She discovered that \(0.6\) of the students preferred playing football. Use mathematical tables to find the reciprocal of \(0.6\text{.}\)
🔗
Answer.
\(0.6 = 6 \times 10^{-1}\)
🔗
Reciprocal of \(0.6\) is \(1.667\)
🔗
🔗
🔗
4.
A farmer can plant \(0.85\) acres of land in one day. How many days will it take the farmer to plant one acre?(Hint: Use the reciprocal of the planting rate.)Use mathematical tables to find your answer.
🔗
Answer.
Number of days by acre is given by \(\frac{1}{0.85}\)
🔗
\(\frac{1}{0.85} = 1.176\)
🔗
🔗
🔗
5.
John can type \(80\) words per minute.
  1. How many minutes does it take him to type one word?
    🔗
    🔗
  2. Based on your answer, how long will it take him to type a \(400\)-word essay?
    🔗
    🔗
🔗
Answer.
  1. \begin{align*} \frac{1}{80} =\amp \frac{1}{8.0 \times 10^{-2}}\\ = \amp 0.1250 \times 10^{-1} \\ =\amp 0.0125 \end{align*}
    🔗
    🔗
  2. \(400 \times 0.0125 = 5\) minutes.
    🔗
    🔗
🔗
🔗
🔗
6.
A shopkeeper has \(96\) apples and packs them in groups of \(8\) apples per pack.
  1. How many packs does the shopkeeper make?
    🔗
    🔗
  2. What is the reciprocal of the number of apples per pack?
    🔗
    🔗
🔗
Answer.
  1. \begin{align*} = \amp 96 \times \frac{1}{8} \\ = \amp 96 \times 0.1250 \\ = \amp 12 \text{ packs} \end{align*}
    🔗
    🔗
  2. \(\displaystyle 0.1250\)
    🔗
    🔗
🔗
🔗
🔗
🔗

Finding reciprocals using a calculator.

🔗

Learner Experience 1.1.6.

1. Working in groups, use a calculator to work out the reciprocal of \(151.6\)
🔗
  1. Press the keys 1, ÷, 1 ,5 ,1, . ,6 in that order.
    🔗
    🔗
  2. Press the key =
    🔗
    🔗
🔗
2. Read the displayed result. What is the reciprocal of \(151.6\) from the calculator?
🔗
3. Work out the reciprocal of each of the following numbers using the calculator:
🔗
  1. \(\displaystyle 0.0038 \)
    🔗
    🔗
  2. \(\displaystyle 0.5498\)
    🔗
    🔗
  3. \(\displaystyle \frac{1}{8}\)
    🔗
    🔗
  4. \(\displaystyle 564\)
    🔗
    🔗
🔗
4. Discuss with other learners how you determine the reciprocal of a number using a calculator.
🔗
🔗

Key Takeaway 1.1.38.

To find reciprocal of a number using a calculator:
🔗
  1. Enter the number into the calculator,
    🔗
    🔗
  2. Press the reciprocal button x⁻¹ or divide \(1\) by the number.
    🔗
    🔗
  3. Read the displayed result.
    🔗
    🔗
🔗
Example: To find the reciprocal of \(7\text{,}\)
🔗
  1. Press the key 1
    🔗
    🔗
  2. Press the key ÷
    🔗
    🔗
  3. Press the key 7
    🔗
    🔗
  4. Press the key =
    🔗
    🔗
🔗
Read the displayed result which is \(0.14285714286\text{.}\) Hence the reciprocal of \(7\) given to \(4\) decimal places is \(0.1429\text{.}\)
🔗
🔗

Example 1.1.39.

Find the reciprocal of the following using a calculator.
🔗
  1. \(\displaystyle 5.6\)
    🔗
    🔗
  2. \(\displaystyle 0.003\)
    🔗
    🔗
  3. \(\displaystyle 12.8\)
    🔗
    🔗
🔗
Solution.
i) To find the reciprocal of \(5.6\text{,}\) press the keys 1, ÷, 5, . then 6.
🔗
Press the key =
🔗
Read the reciprocal of \(5.6\) on the screen of the calculator.
🔗
Hence the reciprocal of \(5.6\) from the calculator is approximately \(0.1786\text{,}\) when rounded off to \(4\) decimal places.
🔗
ii) To calculate the reciprocal of \(0.003\) :
🔗
Press the keys 1,÷, 0, ., 0, 0, then 3.
🔗
Press the key =
🔗
Read the reciprocal of \(0.003\) on the screen of the calculator.
🔗
Therefore, the reciprocal of \(0.003\) from the calculator is: \(333.333333...\text{.}\) Also written as \(333.3333\) when rounded off to \(4\) decimal places.
🔗
iii) To calculate the reciprocal of \(12.8\text{:}\)
🔗
Press the keys 1, ÷, 1, 2, . then 8.
🔗
Press the key =
🔗
Read the reciprocal of \(12.8\) on the screen of the calculator.
🔗
Hence the reciprocal of \(12.8\) from the calculator is: \(0.078125\text{.}\)
🔗
🔗
🔗

Checkpoint 1.1.40. Using a Calculator to Find Reciprocals.

Load the question by clicking the button below.
🔗
🔗

Checkpoint 1.1.41. Finding Reciprocals of Rational and Irrational Numbers.

Load the question by clicking the button below.
🔗
🔗

Exercises Exercise

1.
Find the reciprocal of the following numbers using a calculator:
🔗
  1. \(\displaystyle 8\)
    🔗
    🔗
  2. \(\displaystyle 125\)
    🔗
    🔗
  3. \(\displaystyle 598\)
    🔗
    🔗
  4. \(\displaystyle 8638\)
    🔗
    🔗
  5. \(\displaystyle 8.861\)
    🔗
    🔗
  6. \(\displaystyle 0.00067\)
    🔗
    🔗
  7. \(\displaystyle 0.01467\)
    🔗
    🔗
  8. \(\displaystyle 0.4875\)
    🔗
    🔗
🔗
Answer.
The reciprocals are:
  1. \(\displaystyle 0.125\)
    🔗
    🔗
  2. \(\displaystyle 0.008\)
    🔗
    🔗
  3. \(\displaystyle 0.001672240802675585\)
    🔗
    🔗
  4. \(\displaystyle 0.000115767538782125\)
    🔗
    🔗
  5. \(\displaystyle 0.11285407967498025\)
    🔗
    🔗
  6. \(\displaystyle 1492.537313432835821\)
    🔗
    🔗
  7. \(\displaystyle 68.16632583503749148\)
    🔗
    🔗
  8. \(\displaystyle 2.051282051282051282\)
    🔗
    🔗
🔗
🔗
🔗
2.
A school cafeteria has \(8\) large trays of food to serve equally among the students. To find out how much food each student will get per tray, the cafeteria manager needs to calculate the reciprocal of \(8\text{.}\) Using a calculator, how can the manager find the reciprocal of \(8\text{,}\) and what is the answer?
🔗
Answer.
The manager can find the reciprocal of \(8\) by pressing the keys 1, ÷, 8, then =. The answer displayed on the calculator is \(0.125\text{.}\)
🔗
🔗
🔗
3.
A group of friends has \(5\) bottles of juice to share equally. Use a calculator to determine how much juice each person gets per bottle.
🔗
Answer.
Calculate the reciprocal of \(5\) which is \(0.2\text{.}\)
🔗
🔗
🔗
4.
If a machine completes a task in \(6\) hours, its work rate per hour is the reciprocal of the time. Use a calculator to determine the reciprocal and explain what it represents.
🔗
Answer.
The reciprocal of \(6\) is \(0.1667\) (rounded to four decimal places). This represents the fraction of the task the machine completes in one hour.
🔗
🔗
🔗
5.
A car travels \(12\) kilometers on \(1\) liter of fuel. Use a calculator to find out liters of fuel needed per kilometer.
🔗
Answer.
The reciprocal of \(12\) is \(0.0833\) (rounded to four decimal places). This means the car uses approximately \(0.0833\) liters of fuel per kilometer.
🔗
🔗
🔗
6.
Write down \(3\) numbers and work out their reciprocals using a calculator.
🔗
Answer.
The reciprocals of the numbers is calculated using a calculator by entering \(1\) divided by each number you wrote down.
🔗
🔗
🔗
🔗
🔗

Subsubsection 1.1.2.3 Using Reciprocals

This is application of the concept of reciprocals to perform different types of mathematical operations such as division, solving equations, proportions, rates, and real-life problem-solving.
🔗

Learner Experience 1.1.7.

In groups, find an exercise book and a pen.
🔗
1. Write different numbers (that is positive, negative, decimal numbers and fractions) on the exercise book.
🔗
Example: \(2, -4,0.25,\frac{5}{8}\)
🔗
2. Find the reciprocals of the numbers you have formed.
🔗
Example: Number formed is \(\frac{5}{8}\) its reciprocal is given by \(1 \div \frac{5}{8} = 1 \times \frac{8}{5} = \frac{8}{5}\)
🔗
3. Multiply the original number you formed by its corresponding reciprocal and observe the result.
🔗
Example: \(\frac{5}{8} \times \frac{8}{5}\)
🔗
4. Discuss cases where reciprocals do not exist.
🔗
5. Identify a real world problem where reciprocals are useful.
🔗
6. Share your work with fellow learners.
🔗
🔗

Key Takeaway 1.1.43.

How reciprocals are used in mathematical computations:
🔗
  1. Converting Division into Multiplication
    🔗
    Instead of dividing a number, we multiply by the reciprocal: \(a \div b = a \times \frac{1}{b}\text{.}\)
    🔗
    Example with whole numbers:
    🔗
    \(8 \div 2 = 8 \times \frac{1}{2} = 4\)
    🔗
    Example with fractions:
    🔗
    \begin{align*} \frac{5}{6} \div \frac{2}{3}=\amp \frac{5}{6} \times \frac{3}{2}\\ =\amp \frac{15}{12}\\ =\amp \frac{5}{4} \end{align*}
    🔗
    🔗
  2. Solving Equations
    🔗
    When solving equations where a variable is multiplied by a number, we use the reciprocal to isolate the variable.
    🔗
    Example: Solve for \(x\) in the equation \(3x = 12\)
    🔗
    To solve for \(x\text{,}\) multiply both sides by the reciprocal of \(3\text{.}\)
    🔗
    Therefore, \(x = 12 \times \frac{1}{3} = 4\)
    🔗
    🔗
  3. Working with proportions and ratios while cooking.
    🔗
    Example: If a recipe uses \(\frac{2}{3}\) of a cup of sugar per serving, how many servings can you make with \(4\) cups?
    🔗
    To find the number of servings, compute: \(4 \div \frac{2}{3}\)
    🔗
    \begin{align*} =\amp 4 \times \frac{3}{2}\\ =\amp \frac{12}{2} \\ = \amp \, 6 \text{ servings} \end{align*}
    🔗
    🔗
🔗
🔗

Example 1.1.44.

Kerich is a farmer and has \(\frac{2}{3}\) of an acre of land and wants to divide it into plots of \(\frac{1}{6}\) acre each. How many plots can Kerich make?
🔗
Solution.
To find the number of plots that Kerich can make, we solve it as a division problem.
🔗
Divide \(\frac{2}{3}\) by \(\frac{1}{6}\) and solve it by multiplying \(\frac{2}{3}\) by the reciprocal of \(\frac{1}{6}\) as below:
🔗
\begin{align*} \frac{2}{3} \div \frac{1}{6}=\amp \frac{2}{3} \times \frac{6}{1}\\ \frac{12}{3}=\amp 4 \end{align*}
🔗
Hence Kerich can make \(4\) plots.
🔗
🔗
🔗

Example 1.1.45.

A printing machine can print \(\frac{5}{6}\) of a book page per second. How long will it take to print \(20\) pages?
🔗
Solution.
If \(1\) second = \(\frac{5}{6}\) of a page, then for \(20\) pages, we calculate as follows:
🔗
\begin{align*} \text{ Time } =\amp 20 \div \frac{5}{6}\\ =\amp 20 \times \frac{6}{5} \end{align*}
🔗
Multiplying and simplifying we have: Time = \(\frac{120}{5} = 24\) seconds.
🔗
Thus, the printing machines will take \(24\) seconds to print \(20\) pages.
🔗
🔗
🔗

Example 1.1.46.

Njoki, a business woman invest ksh \(5\,000\) in a business that promises to double his investment every \(1.5\) years. What fraction of her initial investment will she have after \(4.5\) years?
🔗
Solution.
If \(1.5\) years \(= 2\) times the initial investment, how about \(4.5\) years?
🔗
Computing the number of times her investment will yield after \(4.5\) years we have: \(4.5 \div 1.5\)
🔗
\begin{align*} = \amp 4.5 \times \frac{1}{1.5}\\ = \amp 3 \end{align*}
🔗
This means her investment will double \(3\) times.
🔗
Therefore, after \(4.5\) years her investment will be \(5000 \times 2\times2\times2\)
🔗
\begin{align*} = \amp 5000 \times 8\\ = \amp 40\,000 \end{align*}
🔗
Illustration
🔗
Representing the number of years with their corresponding double investment in ksh makes it easier to understand the above computations.
🔗
Remember at the start of a period we will represent it with a \(0\text{.}\)
🔗
Years: \(0\) \(\rightarrow\) \(1.5\) \(\rightarrow\) \(3\) \(\rightarrow\) \(4.5\)
Shillings: \(5\,000\) \(\rightarrow\) \(10\,000\) \(\rightarrow\) \(20\,000\) \(\rightarrow\) \(40\,000\)
Hence Njoki will have ksh \(40\,000\) after \(4.5\) years.
🔗
🔗
🔗

Example 1.1.47.

Okoth runs \(3 \frac{1}{4}\) miles in \(\frac{1}{2}\) an hour. What is her speed in miles per hour?
🔗
Solution.
To find speed we divide distance by time such that: speed = \(\frac{\text{ distance}}{\text{ time}}\text{.}\)
🔗
First, we need to convert the fraction \(3\frac{1}{4}\) into an improper fraction which is \(\frac{13}{4}\)
🔗
Hence, speed = \(3\frac{1}{4} \div \frac{1}{2} = \frac{13}{4} \div \frac{1}{2}\)
🔗
Multiply \(\frac{13}{4}\) by the reciprocal of \(\frac{1}{2}\text{.}\)
🔗
\(\frac{13}{4} \times \frac{2}{1} = \frac{26}{4}\)
🔗
Simplifying \(\frac{26}{4}\) we get \(\frac{13}{2}\) which is \(= 6 \frac{1}{2}\text{.}\)
🔗
🔗
🔗

Checkpoint 1.1.48.

🔗

Checkpoint 1.1.49.

🔗

Exercises Exercises

1.
Solve the following using reciprocals:
🔗
  1. \(\displaystyle 6 \div \frac{2}{3}\)
    🔗
    🔗
  2. \(\displaystyle 4x = 12\)
    🔗
    🔗
  3. \(\displaystyle \frac{3}{5} \times x = 9\)
    🔗
    🔗
  4. \(\displaystyle 0.2 \div 4.5\)
    🔗
    🔗
🔗
Answer.
  1. \(\displaystyle 6 \times \frac{3}{2} = 9\)
    🔗
    🔗
  2. \(\displaystyle x = 12 \times \frac{1}{4} = 3\)
    🔗
    🔗
  3. \(\displaystyle x = 9 \times \frac{5}{3} = 15\)
    🔗
    🔗
  4. \(\displaystyle 0.2 \times \frac{1}{4.5} = \frac{2}{45} = 0.0444\)
    🔗
    🔗
🔗
🔗
🔗
2.
Solve: \(\left( \frac{4}{7} \div \frac{2}{5}\right)\) \(\times\) \(\left(\frac{3}{8} \div \frac{9}{16}\right)\)
🔗
Answer.
\begin{align*} =\amp \left( \frac{4}{7} \times \frac{5}{2}\right) \times \left(\frac{3}{8} \times \frac{16}{9}\right)\\ =\amp \frac{20}{14} \times \frac{48}{72} \\ =\amp \frac{10}{7} \times \frac{2}{3} \\ =\amp \frac{20}{21} \end{align*}
🔗
🔗
🔗
3.
If a car travels at \(80\) km/h, to find out the time taken per km we can use reciprocal.
🔗
Answer.
Time per km = \(\frac{1}{80}\) hours per km.
🔗
🔗
🔗
4.
If a factory produces \(300\) items in \(5\) hours, what is the production rate per hour?
🔗
Solution.
To determine how much time is needed per item, we use reciprocal:
🔗
\(\frac{300}{5} = 60\) items per hour.
🔗
🔗
🔗
5.
If \(x \times \frac{3}{8} = \frac{5}{12}\text{,}\) find the value of \(x\) using reciprocals.
🔗
Answer.
\(x = \frac{5}{12} \times \frac{8}{3} = \frac{40}{36} = \frac{10}{9}\)
🔗
🔗
🔗
6.
A rope of length \(18\) m is cut into pieces each measuring \(\frac{3}{4}\) m. How many pieces are obtained?
🔗
Answer.
\(18 \div \frac{3}{4} = 18 \times \frac{4}{3} = 24\) pieces.
🔗
🔗
🔗
7.
A hiker is walking \(24\) km and takes breaks every \(\frac{2}{3}\) km. How many breaks will he take by the end of the journey? If he changes his break interval to \(1\frac{1}{3}\) km, how does the number of breaks change?
🔗
Answer.
Number of breaks at \(\frac{2}{3}\) km intervals: \(24 \times \frac{3}{2} = 36\) breaks.
🔗
Number of breaks at \(1\frac{1}{3}\) km intervals: \(= 24 \times \frac{3}{4} = 18\) breaks.
🔗
Changing the break interval to \(1\frac{1}{3}\) km reduces the number of breaks from \(36\) to \(18\text{.}\)
🔗
🔗
🔗
8.
A road is \(12.6\) km long and is divided into equal segments of \(0.35\) km for maintenance. How many segments are there?
🔗
Answer.
\(12.6 \times \frac{1}{0.35} = 36\) segments.
🔗
🔗
🔗
🔗
🔗
🔗
PrevTopNext