{"id":9884,"date":"2026-07-03T17:39:42","date_gmt":"2026-07-03T17:39:42","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=9884"},"modified":"2026-07-03T17:39:42","modified_gmt":"2026-07-03T17:39:42","slug":"direct-inverse-proportion","status":"publish","type":"post","link":"https:\/\/kapdec.com\/help\/direct-inverse-proportion\/","title":{"rendered":"Direct & Inverse Proportion"},"content":{"rendered":"
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KAPDEC® | Elite STEM Learning Platform | https:\/\/kapdec.com<\/a><\/p>\n

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Unit: <\/strong>Factorization Of Expressions<\/strong><\/h2>\n

Chapter: <\/strong>Direct & Inverse Proportion<\/strong><\/h3>\n

Reference: – What is Proportion, Direct Proportion Definition, Direct Proportion Formula (y = kx), Constant of Proportionality (k), Real-Life Examples of Direct Proportion, Graph of Direct Proportion, Inverse Proportion Definition, Inverse Proportion Formula (y = k\/x), Real-Life Examples of Inverse Proportion, Graph of Inverse Proportion, Solving Proportion Problems, Solved Examples, Odd-One-Out Problems, Common Mistakes<\/em><\/p>\n

After studying this chapter, you should be able to understand:<\/strong><\/p>\n

    \n
  • What is Direct Proportion and How to Identify It<\/em><\/li>\n
  • What is Inverse Proportion and How to Identify It<\/em><\/li>\n
  • How to Find the Constant of Proportionality<\/em><\/li>\n
  • How to Solve Direct and Inverse Proportion Problems<\/em><\/li>\n
  • How to Recognize the Graphs of Direct and Inverse Proportion<\/em><\/li>\n<\/ul>\n

    Introduction to Direct and Inverse Proportion<\/strong><\/p>\n

    Definition<\/u><\/strong><\/p>\n

    Proportion describes the relationship between two quantities. When two quantities change in relation to each other, they are either in direct proportion or inverse proportion.<\/p>\n

    When we study proportion, we essentially ask:<\/p>\n

    “As one quantity increases, what happens to the other quantity? Does it also increase, or does it decrease?”<\/p>\n

    The answer determines whether the relationship is direct or inverse.<\/p>\n

    Importance of Proportion<\/u><\/strong><\/p>\n

      \n
    • Used in everyday situations (speed, time, cost, recipes)<\/li>\n
    • Essential for scaling and resizing (maps, blueprints)<\/li>\n
    • Helps solve problems without complex equations<\/li>\n
    • Foundation for ratio and percentage concepts<\/li>\n
    • Used in science (density, pressure, gas laws)<\/li>\n<\/ul>\n

      Example \u2013 Direct Proportion:\u00a0The more hours you work, the more money you earn. As one increases, the other increases.<\/p>\n

      Example \u2013 Inverse Proportion:<\/strong>\u00a0The faster you drive, the less time a trip takes. As one increases, the other decreases.<\/p>\n

      Subtopics<\/u><\/strong><\/p>\n

      1. Direct Proportion<\/strong><\/p>\n

      Two quantities are in\u00a0direct proportion\u00a0when they increase or decrease together at the same rate. If one doubles, the other doubles. If one triples, the other triples.<\/p>\n

      Key Property:\u00a0The ratio between the two quantities is always constant.<\/p>\n

      Formula:\u00a0y = kx<\/p>\n

      Where y and x are the two quantities, and k is the\u00a0constant of proportionality.<\/p>\n

      Constant of Proportionality (k):\u00a0k = y\/x (the same value for all pairs)<\/p>\n

      Examples of Direct Proportion:<\/strong><\/p>\n

        \n
      • Cost of apples and number of apples (cost = price per apple \u00d7 number)<\/li>\n
      • Distance travelled and time at constant speed (distance = speed \u00d7 time)<\/li>\n
      • Weight and mass (weight = gravity \u00d7 mass)<\/li>\n
      • Circumference and diameter of a circle (C = \u03c0 \u00d7 d)<\/li>\n
      • Amount of ingredients and number of servings in a recipe<\/li>\n<\/ul>\n

        How to Solve Direct Proportion Problems:<\/strong><\/p>\n

        Step 1: Identify the two quantities and write the relationship y = kx<\/p>\n

        Step 2: Use the given pair of values to find k<\/p>\n

        Step 3: Use k to find the unknown value<\/p>\n

        Example 1:<\/strong>\u00a0If 3 pencils cost $6, how much do 8 pencils cost?<\/p>\n

        Here cost \u221d number of pencils. Let c = cost, n = number. c = k \u00d7 n<\/p>\n

        k = c\/n = 6\/3 = 2 (cost per pencil = $2)<\/p>\n

        For 8 pencils: c = 2 \u00d7 8 = $16<\/p>\n

        Answer:<\/strong>\u00a0$16<\/p>\n

        Example 2:<\/strong>\u00a0A car travels 120 miles in 2 hours. How far will it travel in 5 hours at the same speed?<\/p>\n

        Distance \u221d time. d = k \u00d7 t<\/p>\n

        k = d\/t = 120\/2 = 60 miles per hour<\/p>\n

        In 5 hours: d = 60 \u00d7 5 = 300 miles<\/p>\n

        Answer:<\/strong>\u00a0300 miles<\/p>\n

        Graph of Direct Proportion:\u00a0A straight line passing through the origin (0,0). As x increases, y increases.<\/p>\n

        2. Inverse Proportion<\/strong><\/p>\n

        Two quantities are in\u00a0inverse proportion\u00a0when one increases and the other decreases at the same rate. If one doubles, the other halves. If one triples, the other becomes one-third.<\/p>\n

        Key Property:\u00a0The product of the two quantities is always constant.<\/p>\n

        Formula:\u00a0y = k\/x (or xy = k)<\/p>\n

        Constant of Proportionality (k):\u00a0k = x \u00d7 y (the same value for all pairs)<\/p>\n

        Examples of Inverse Proportion:<\/strong><\/p>\n

          \n
        • Speed and time for a fixed distance (faster speed = less time)<\/li>\n
        • Number of workers and time to complete a job (more workers = less time)<\/li>\n
        • Price per item and number purchased with a fixed budget (higher price = fewer items)<\/li>\n
        • Pressure and volume of a gas at constant temperature (Boyle’s Law)<\/li>\n<\/ul>\n

          How to Solve Inverse Proportion Problems:<\/strong><\/p>\n

          Step 1: Identify the two quantities and write the relationship xy = k<\/p>\n

          Step 2: Use the given pair of values to find k<\/p>\n

          Step 3: Use k to find the unknown value<\/p>\n

          Example 1:<\/strong>\u00a06 workers can build a wall in 10 days. How many days will 15 workers take to build the same wall?<\/p>\n

          Workers \u221d 1\/time, so workers \u00d7 days = k<\/p>\n

          k = 6 \u00d7 10 = 60 (total worker-days)<\/p>\n

          For 15 workers: 15 \u00d7 d = 60 \u2192 d = 60\/15 = 4 days<\/p>\n

          Answer:<\/strong>\u00a04 days<\/p>\n

          Example 2:<\/strong>\u00a0A car traveling at 50 mph takes 6 hours to reach a destination. How long will it take at 75 mph?<\/p>\n

          Speed \u00d7 time = k<\/p>\n

          k = 50 \u00d7 6 = 300<\/p>\n

          At 75 mph: 75 \u00d7 t = 300 \u2192 t = 300\/75 = 4 hours<\/p>\n

          Answer:<\/strong>\u00a04 hours<\/p>\n

          Graph of Inverse Proportion:<\/strong>\u00a0A curve (hyperbola) that never touches the axes. As x increases, y decreases.<\/p>\n

          4. Word Problems \u2013 Step by Step<\/strong><\/p>\n

          Direct Proportion Word Problem Strategy:<\/strong><\/p>\n

          Step 1: Identify the two variables and write the proportion statement (y \u221d x)<\/p>\n

          Step 2: Set up the equation y = kx<\/p>\n

          Step 3: Find k using the given pair<\/p>\n

          Step 4: Substitute the new value to find the unknown<\/p>\n

          Inverse Proportion Word Problem Strategy:<\/strong><\/p>\n

          Step 1: Identify the two variables and write the proportion statement (y \u221d 1\/x)<\/p>\n

          Step 2: Set up the equation xy = k<\/p>\n

          Step 3: Find k using the given pair<\/p>\n

          Step 4: Substitute the new value to find the unknown<\/p>\n

          5. Direct Proportion in Tables<\/strong><\/p>\n

          In a table showing direct proportion, the ratio y\/x is the same for all pairs.<\/p>\n

          Example \u2013 Direct Proportion Table:<\/strong><\/p>\n

          \n\n\n\n\n\n
          \n

          x<\/p>\n<\/td>\n

          		\n

          2<\/p>\n<\/td>\n

          		\n

          4<\/p>\n<\/td>\n

          		\n

          6<\/p>\n<\/td>\n

          		\n

          8<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

          \n

          y<\/p>\n<\/td>\n

          		\n

          6<\/p>\n<\/td>\n

          		\n

          12<\/p>\n<\/td>\n

          		\n

          18<\/p>\n<\/td>\n

          		\n

          24<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

          Source: Kapdec.com<\/p>\n<\/div>\n

          Check ratios: 6\/2 = 3, 12\/4 = 3, 18\/6 = 3, 24\/8 = 3 \u2713 Constant<\/p>\n

          Example \u2013 Not Direct Proportion:<\/strong><\/p>\n

          \n\n\n\n\n\n
          \n

          x<\/p>\n<\/td>\n

          		\n

          2<\/p>\n<\/td>\n

          		\n

          4<\/p>\n<\/td>\n

          		\n

          6<\/p>\n<\/td>\n

          		\n

          8<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

          \n

          y<\/p>\n<\/td>\n

          		\n

          5<\/p>\n<\/td>\n

          		\n

          9<\/p>\n<\/td>\n

          		\n

          13<\/p>\n<\/td>\n

          		\n

          17<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

          Source: Kapdec.com<\/p>\n<\/div>\n

          Ratios: 5\/2 = 2.5, 9\/4 = 2.25 \u2013 not constant<\/p>\n

          6. Inverse Proportion in Tables<\/strong><\/p>\n

          In a table showing inverse proportion, the product xy is the same for all pairs.<\/p>\n

          Example \u2013 Inverse Proportion Table:<\/p>\n

          \n\n\n\n\n\n
          \n

          x<\/p>\n<\/td>\n

          		\n

          2<\/p>\n<\/td>\n

          		\n

          4<\/p>\n<\/td>\n

          		\n

          5<\/p>\n<\/td>\n

          		\n

          10<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

          \n

          y<\/p>\n<\/td>\n

          		\n

          30<\/p>\n<\/td>\n

          		\n

          15<\/p>\n<\/td>\n

          		\n

          12<\/p>\n<\/td>\n

          		\n

          6<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

          Source: Kapdec.com<\/p>\n<\/div>\n

          Check products: 2\u00d730=60, 4\u00d715=60, 5\u00d712=60, 10\u00d76=60 \u2713 Constant<\/p>\n

          \u00a0<\/p>\n

          Solved Examples<\/strong><\/p>\n

          Example 1 \u2013 Direct Proportion:<\/strong>\u00a0If 4 kg of rice costs $20, how much do 7 kg cost?<\/p>\n

          Solution:<\/strong>\u00a0Cost \u221d weight \u2192 C = k \u00d7 w<\/p>\n

          k = C\/w = 20\/4 = 5<\/p>\n

          For 7 kg: C = 5 \u00d7 7 = $35<\/p>\n

          Answer:<\/strong>\u00a0$35<\/p>\n

          \u00a0<\/p>\n

          Example 2 \u2013 Inverse Proportion:<\/strong>\u00a05 taps can fill a tank in 12 minutes. How long will 8 taps take?<\/p>\n

          Solution:<\/strong>\u00a0Taps \u00d7 time = k<\/p>\n

          k = 5 \u00d7 12 = 60<\/p>\n

          For 8 taps: 8 \u00d7 t = 60 \u2192 t = 60\/8 = 7.5 minutes<\/p>\n

          Answer:<\/strong>\u00a07.5 minutes<\/p>\n

          \u00a0<\/p>\n

          Example 3 \u2013 Find k in Direct Proportion:<\/strong>\u00a0y varies directly with x. When x = 3, y = 21. Find y when x = 8.<\/p>\n

          Solution:<\/strong>\u00a0y = kx \u2192 k = y\/x = 21\/3 = 7<\/p>\n

          When x = 8: y = 7 \u00d7 8 = 56<\/p>\n

          Answer:<\/strong>\u00a0y = 56<\/p>\n

          \u00a0<\/p>\n

          Example 4 \u2013 Find k in Inverse Proportion:<\/strong>\u00a0y varies inversely with x. When x = 4, y = 9. Find y when x = 6.<\/p>\n

          Solution:<\/strong>\u00a0xy = k \u2192 k = 4 \u00d7 9 = 36<\/p>\n

          When x = 6: 6 \u00d7 y = 36 \u2192 y = 36\/6 = 6<\/p>\n

          Answer:<\/strong>\u00a0y = 6<\/p>\n

          \u00a0<\/p>\n

          Common Mistakes to Avoid<\/u><\/strong><\/p>\n

          Mistake 1 \u2013 Confusing direct and inverse proportion<\/strong>
          \nThinking “more workers means more time” is wrong. More workers actually mean less time (inverse).
          \nCorrect understanding: Identify whether the quantities move together (direct) or opposite (inverse).<\/p>\n

          Mistake 2 \u2013 Using the wrong formula<\/strong>
          \nUsing y = kx for inverse proportion or xy = k for direct proportion leads to wrong answers.
          \nCorrect understanding: Direct \u2192 y\/x = k; Inverse \u2192 xy = k.<\/p>\n

          Mistake 3 \u2013 Forgetting to find k first<\/strong>
          \nTrying to solve proportion problems without finding the constant of proportionality leads to errors.
          \nCorrect understanding: Always find k using the given pair before solving for the unknown.<\/p>\n

          Mistake 4 \u2013 Not checking if the graph passes through the origin<\/strong>
          \nA direct proportion graph must pass through (0,0). If it doesn’t, it is not direct proportion.
          \nCorrect understanding: y = kx always gives (0,0). If there is a fixed starting value, it is not direct proportion.<\/p>\n

          Mistake 5 \u2013 Assuming all increasing relationships are direct<\/strong>
          \nA relationship can be increasing but not proportional (like y = 2x + 1).
          \nCorrect understanding: Direct proportion requires y\/x to be constant and the line to pass through the origin.<\/p>\n

          Mistake 6 \u2013 Dividing instead of multiplying for inverse proportion<\/strong>
          \nIn inverse proportion, if x doubles, y halves (divide by 2), not subtract something.
          \nCorrect understanding: Use the formula xy = k to find new values.<\/p>\n

          \u00a0<\/p>\n

          Quick Reference Summary<\/u><\/strong><\/p>\n

          Direct Proportion:<\/strong>\u00a0y = kx (y\/x = k constant)
          \nAs x increases, y increases
          \nGraph: straight line through origin (0,0)<\/p>\n

          Inverse Proportion:<\/strong>\u00a0y = k\/x (xy = k constant)
          \nAs x increases, y decreases
          \nGraph: curve (hyperbola)<\/p>\n

          Constant of Proportionality (k):<\/strong>
          \nDirect: k = y\/x
          \nInverse: k = xy<\/p>\n

          To Solve Direct Proportion:<\/strong>\u00a0Find k = y\/x, then y = k \u00d7 new x<\/p>\n

          To Solve Inverse Proportion:<\/strong>\u00a0Find k = x \u00d7 y, then new y = k \u00f7 new x<\/p>\n

          Real-Life Examples:<\/strong>
          \nDirect: cost and quantity, distance and time (constant speed)
          \nInverse: speed and time (fixed distance), workers and time (fixed job)<\/p>\n

          \u00a0<\/p>\n

          <\/p>\n