{"id":1274,"date":"2026-01-26T17:49:58","date_gmt":"2026-01-26T07:49:58","guid":{"rendered":"https:\/\/archive4ones.com\/2ndstudy\/?p=1274"},"modified":"2026-01-27T12:15:35","modified_gmt":"2026-01-27T02:15:35","slug":"year12-math-1-2-1-measurement-scales-and-data-compound-interest-comparison-problem","status":"publish","type":"post","link":"https:\/\/archive4ones.com\/2ndstudy\/?p=1274","title":{"rendered":"Year12- MATH-1-3-1 Measurement, Scales, and Data. & Compound Interest comparison problem"},"content":{"rendered":"\n

<\/p>\n\n\n\n

Unit 3:<\/strong><\/p>\n\n\n\n

This moves us into Unit 3: Measurement, Scales, and Data<\/strong>. This unit is the bread and butter of the PSMT (Problem-Solving and Modelling Task)<\/strong>, which is that big report you have to write in Year 12.<\/p>\n\n\n\n

In this topic, you aren’t just doing math; you\u2019re acting like a project manager or a builder.<\/p>\n\n\n\n

\n\n\n\n

The Practice Problem: “The Deck Extension”<\/h3>\n\n\n\n

The Scenario:<\/strong> You are looking at a house plan for a new outdoor deck. On the paper plan, the deck is drawn as a rectangle. You need to calculate the cost of the timber flooring.<\/p>\n\n\n\n

The Data:<\/strong><\/p>\n\n\n\n

    \n
  • Scale on Plan:<\/strong> 1<\/mn>:<\/mo>50<\/mn><\/mrow>1:50<\/annotation><\/semantics><\/math><\/li>\n\n\n\n
  • Plan Dimensions:<\/strong> The deck measures 12<\/mn> cm<\/mtext><\/mrow>12 \\text{ cm}<\/annotation><\/semantics><\/math> long and 8.4<\/mn> cm<\/mtext><\/mrow>8.4 \\text{ cm}<\/annotation><\/semantics><\/math> wide on the paper.<\/li>\n\n\n\n
  • Flooring Cost:<\/strong> The timber decking costs $<\/mi>85<\/mn> per square metre (m<\/mtext>2<\/mn><\/msup>)<\/mtext><\/mrow>\\$85 \\text{ per square metre (m}^2\\text{)}<\/annotation><\/semantics><\/math>.<\/li>\n<\/ul>\n\n\n\n

    Your Tasks:<\/strong><\/p>\n\n\n\n

      \n
    1. Calculate<\/strong> the actual real-life length and width of the deck in metres.<\/li>\n\n\n\n
    2. Calculate<\/strong> the total real-life area of the deck in m<\/mtext>2<\/mn><\/msup>\\text{m}^2<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n
    3. Calculate<\/strong> the total cost of the timber required for the deck.<\/li>\n<\/ol>\n\n\n\n
      \n\n\n\n

      Worked Solution (The “QCAA Way”)<\/h3>\n\n\n\n

      Step 1: Convert Plan to Real Life<\/h3>\n\n\n\n

      A scale of 1<\/mn>:<\/mo>50<\/mn><\/mrow>1:50<\/annotation><\/semantics><\/math> means 1<\/mn> unit<\/mtext><\/mrow>1 \\text{ unit}<\/annotation><\/semantics><\/math> on the page equals 50<\/mn> units<\/mtext><\/mrow>50 \\text{ units}<\/annotation><\/semantics><\/math> in real life.<\/p>\n\n\n\n
        \n
      • Actual Length:<\/strong>12<\/mn> cm<\/mtext>\u00d7<\/mo>50<\/mn>=<\/mo>600<\/mn> cm<\/mtext><\/mrow>12 \\text{ cm} \\times 50 = 600 \\text{ cm}<\/annotation><\/semantics><\/math> Convert to metres: 600<\/mn>\u00f7<\/mo>100<\/mn>=<\/mo>\ud835\udfd4<\/mn> m<\/mtext><\/mrow><\/mrow>600 \\div 100 = \\mathbf{6 \\text{ m}}<\/annotation><\/semantics><\/math><\/li>\n\n\n\n
      • Actual Width:<\/strong>8.4<\/mn> cm<\/mtext>\u00d7<\/mo>50<\/mn>=<\/mo>420<\/mn> cm<\/mtext><\/mrow>8.4 \\text{ cm} \\times 50 = 420 \\text{ cm}<\/annotation><\/semantics><\/math> Convert to metres: 420<\/mn>\u00f7<\/mo>100<\/mn>=<\/mo>\ud835\udfd2.\ud835\udfd0<\/mn> m<\/mtext><\/mrow><\/mrow>420 \\div 100 = \\mathbf{4.2 \\text{ m}}<\/annotation><\/semantics><\/math><\/li>\n<\/ul>\n\n\n\n

        Step 2: Find the Area<\/h3>\n\n\n\n

        Now that we have the real-world dimensions in metres, we find the area (A<\/mi>=<\/mo>L<\/mi>\u00d7<\/mo>W<\/mi><\/mrow>A = L \\times W<\/annotation><\/semantics><\/math>):<\/p>\n\n\n\n

        A<\/mi>=<\/mo>6<\/mn>\u00d7<\/mo>4.2<\/mn><\/mrow>A = 6 \\times 4.2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n

        A<\/mi>=<\/mo>\ud835\udfd0\ud835\udfd3.\ud835\udfd0<\/mn> m<\/mtext>\ud835\udfd0<\/mn><\/msup><\/mrow><\/mrow>A = \\mathbf{25.2 \\text{ m}^2}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n

        Step 3: Calculate the Total Cost<\/h3>\n\n\n\n

        C<\/mi>o<\/mi>s<\/mi>t<\/mi>=<\/mo>A<\/mi>r<\/mi>e<\/mi>a<\/mi>\u00d7<\/mo>Rate per m<\/mtext>2<\/mn><\/msup><\/mrow>Cost = Area \\times \\text{Rate per m}^2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n

        C<\/mi>o<\/mi>s<\/mi>t<\/mi>=<\/mo>25.2<\/mn>\u00d7<\/mo>85<\/mn><\/mrow>Cost = 25.2 \\times 85<\/annotation><\/semantics><\/math><\/p>\n\n\n\n

        Total Cost = <\/strong>$<\/mi>2,142<\/mn><\/mrow>\\$2,142<\/annotation><\/semantics><\/math><\/p>\n\n\n\n
        \n\n\n\n

        Essential Study Tips for Unit 3<\/h3>\n\n\n\n

        1. The “Square Rule” Trap<\/h3>\n\n\n\n

        One of the most common mistakes in Year 12 exams is trying to convert area<\/em> by the same scale as length<\/em>.<\/p>\n\n\n\n

          \n
        • Wrong:<\/strong> “The area on paper is 100.8<\/mn> cm<\/mtext>2<\/mn><\/msup><\/mrow>100.8 \\text{ cm}^2<\/annotation><\/semantics><\/math>, so I’ll just multiply by 50<\/mn>50<\/annotation><\/semantics><\/math>.” (This gives you the wrong answer!)<\/li>\n\n\n\n
        • Right:<\/strong> Always convert the individual side lengths<\/strong> to metres first, then<\/em> multiply them to get the area. It prevents a world of pain.<\/li>\n<\/ul>\n\n\n\n

          2. Units, Units, Units!<\/h3>\n\n\n\n

          In Essential Maths, you will constantly jump between mm<\/mtext>\\text{mm}<\/annotation><\/semantics><\/math>, cm<\/mtext>\\text{cm}<\/annotation><\/semantics><\/math>, m<\/mtext>\\text{m}<\/annotation><\/semantics><\/math>, and km<\/mtext>\\text{km}<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n
            \n
          • Check:<\/strong> Did you divide by 100<\/mn>100<\/annotation><\/semantics><\/math> to get metres?<\/li>\n\n\n\n
          • Check:<\/strong> Does the answer look right? A 6<\/mn> metre<\/mtext><\/mrow>6 \\text{ metre}<\/annotation><\/semantics><\/math> deck is reasonable. A 600<\/mn> metre<\/mtext><\/mrow>600 \\text{ metre}<\/annotation><\/semantics><\/math> deck is a runway for a Boeing 747.<\/li>\n<\/ul>\n\n\n\n

            3. The “Wastage” Factor (Complex Unfamiliar)<\/h3>\n\n\n\n

            In a real QCAA exam, they might add a “Complex” twist: “Allow 10% extra for timber wastage.”<\/em> * To solve this, you would take your area (25.2<\/mn>25.2<\/annotation><\/semantics><\/math>) and multiply by 1.10<\/mn>1.10<\/annotation><\/semantics><\/math> before calculating the cost.<\/p>\n\n\n\n

            ****************************************************************************<\/p>\n\n\n\n

            ****************************************************************************<\/p>\n\n\n\n

            Unit 4:<\/strong><\/p>\n\n\n\n

            Since Unit 4: Loans and Interest<\/strong> is a massive part of the Year 12 Essential Maths course (and a very handy life skill), let\u2019s tackle a Compound Interest<\/strong> comparison problem.<\/p>\n\n\n\n

            This type of question is a classic “Complex Familiar” task you might see in an exam.<\/p>\n\n\n\n

            \n\n\n\n

            The Practice Problem: “The Car Fund”<\/h3>\n\n\n\n

            >> Problem 1:<\/strong><\/p>\n\n\n\n

            The Scenario:<\/strong> Alex has just finished Year 12 and wants to save $5,000<\/strong> for a second-hand car. They have $4,000<\/strong> to invest right now and plan to leave it in the bank for 3 years<\/strong>. Alex is comparing two different savings accounts:<\/p>\n\n\n\n

              \n
            • Account A:<\/strong> Offers 4.8% p.a.<\/strong> (per annum) Simple Interest<\/strong>.<\/li>\n\n\n\n
            • Account B:<\/strong> Offers 4.5% p.a.<\/strong> Compound Interest<\/strong>, compounded annually<\/strong>.<\/li>\n<\/ul>\n\n\n\n

              Your Tasks:<\/strong><\/p>\n\n\n\n

                \n
              1. Solve:<\/strong> Calculate the total amount Alex will have in Account A<\/strong> after 3 years.<\/li>\n\n\n\n
              2. Solve:<\/strong> Calculate the total amount Alex will have in Account B<\/strong> after 3 years.<\/li>\n\n\n\n
              3. Evaluate:<\/strong> Which account should Alex choose to get closer to their $5,000 goal?<\/li>\n<\/ol>\n\n\n\n
                \n\n\n\n

                Worked Solution (The “QCAA Way”)<\/h3>\n\n\n\n

                Part 1: Account A (Simple Interest)<\/h3>\n\n\n\n

                From your QCAA formula sheet, the formula for Simple Interest is:<\/p>\n\n\n\n

                I<\/mi>=<\/mo>P<\/mi>r<\/mi>t<\/mi><\/mrow>I = Prt<\/annotation><\/semantics><\/math><\/p>\n\n\n\n

                Where P<\/mi>=<\/mo>4000<\/mn><\/mrow>P = 4000<\/annotation><\/semantics><\/math>, r<\/mi>=<\/mo>0.048<\/mn><\/mrow>r = 0.048<\/annotation><\/semantics><\/math> (the decimal of 4.8%), and t<\/mi>=<\/mo>3<\/mn><\/mrow>t = 3<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n
                  \n
                1. I<\/mi>=<\/mo>4000<\/mn>\u00d7<\/mo>0.048<\/mn>\u00d7<\/mo>3<\/mn><\/mrow>I = 4000 \\times 0.048 \\times 3<\/annotation><\/semantics><\/math><\/li>\n\n\n\n
                2. I<\/mi>=<\/mo>576<\/mn><\/mrow>I = 576<\/annotation><\/semantics><\/math><\/li>\n\n\n\n
                3. Total Amount<\/strong> =<\/mo>P<\/mi>+<\/mo>I<\/mi>=<\/mo>4000<\/mn>+<\/mo>576<\/mn>=<\/mo>$<\/mi>\ud835\udfd2,\ud835\udfd3\ud835\udfd5\ud835\udfd4<\/mn><\/mrow><\/mrow>= P + I = 4000 + 576 = \\mathbf{\\$4,576}<\/annotation><\/semantics><\/math><\/li>\n<\/ol>\n\n\n\n

                  Part 2: Account B (Compound Interest)<\/h3>\n\n\n\n

                  The formula for the total amount (A<\/mi>A<\/annotation><\/semantics><\/math>) in Compound Interest is:<\/p>\n\n\n\n

                  A<\/mi>=<\/mo>P<\/mi>(<\/mo>1<\/mn>+<\/mo>r<\/mi>)<\/mo>n<\/mi><\/msup><\/mrow>A = P(1 + r)^n<\/annotation><\/semantics><\/math><\/p>\n\n\n\n

                  Where P<\/mi>=<\/mo>4000<\/mn><\/mrow>P = 4000<\/annotation><\/semantics><\/math>, r<\/mi>=<\/mo>0.045<\/mn><\/mrow>r = 0.045<\/annotation><\/semantics><\/math>, and n<\/mi>=<\/mo>3<\/mn><\/mrow>n = 3<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n
                    \n
                  1. A<\/mi>=<\/mo>4000<\/mn>(<\/mo>1<\/mn>+<\/mo>0.045<\/mn>)<\/mo>3<\/mn><\/msup><\/mrow>A = 4000(1 + 0.045)^3<\/annotation><\/semantics><\/math><\/li>\n\n\n\n
                  2. A<\/mi>=<\/mo>4000<\/mn>(<\/mo>1.045<\/mn>)<\/mo>3<\/mn><\/msup><\/mrow>A = 4000(1.045)^3<\/annotation><\/semantics><\/math><\/li>\n\n\n\n
                  3. A<\/mi>=<\/mo>4000<\/mn>\u00d7<\/mo>1.141166…<\/mn><\/mrow>A = 4000 \\times 1.141166…<\/annotation><\/semantics><\/math><\/li>\n\n\n\n
                  4. Total Amount<\/strong> =<\/mo>$<\/mi>\ud835\udfd2,\ud835\udfd3\ud835\udfd4\ud835\udfd2.\ud835\udfd4\ud835\udfd4<\/mn><\/mrow><\/mrow>= \\mathbf{\\$4,564.66}<\/annotation><\/semantics><\/math> (rounded to 2 decimal places)<\/li>\n<\/ol>\n\n\n\n

                    Part 3: Evaluation & Reasonableness<\/h3>\n\n\n\n
                      \n
                    • Comparison:<\/strong> Account A yields $4,576<\/strong>, while Account B yields $4,564.66<\/strong>.<\/li>\n\n\n\n
                    • Conclusion:<\/strong> Even though Account B’s interest “compounds,” the higher interest rate of Account A (4.8% vs 4.5%) makes it the better choice over this short 3-year period.<\/li>\n\n\n\n
                    • Reasonableness Check:<\/strong> Alex still hasn’t reached the $5,000 goal. They are about $424 short. Alex might need to find an account with a higher rate or save for another year.<\/li>\n<\/ul>\n\n\n\n
                      \n\n\n\n

                      Study Tip for this Topic<\/h3>\n\n\n\n

                      In Essential Maths, they love to change the compounding period<\/strong>. If Account B compounded monthly<\/strong> instead of annually, you would have to:<\/p>\n\n\n\n

                        \n
                      1. Divide the interest rate by 12 (r<\/mi>\u00f7<\/mo>12<\/mn><\/mrow>r \\div 12<\/annotation><\/semantics><\/math>).<\/li>\n\n\n\n
                      2. Multiply the number of years by 12 (n<\/mi>\u00d7<\/mo>12<\/mn><\/mrow>n \\times 12<\/annotation><\/semantics><\/math>).<\/li>\n<\/ol>\n\n\n\n

                        **************************************************************************<\/p>\n\n\n\n

                        >> Problem 2:<\/strong>
                        <\/p>\n\n\n\n

                        That is a great idea. In the Queensland Essential Mathematics syllabus, being able to adjust for different compounding periods is a key skill for Unit 4, Topic 3: Loans and Investments<\/strong>.<\/p>\n\n\n\n

                        Here is the modified challenge and the step-by-step “QCAA style” solution.<\/p>\n\n\n\n

                        \n\n\n\n

                        The Challenge: Monthly Compounding<\/h3>\n\n\n\n

                        The Scenario:<\/strong> Alex still has $4,000<\/strong> to invest for 3 years<\/strong>.<\/p>\n\n\n\n

                          \n
                        • Account A (Reminder):<\/strong> 4.8% p.a. Simple Interest (Total = $4,576<\/strong>)<\/li>\n\n\n\n
                        • Account B (Modified):<\/strong> 4.5% p.a. Compound Interest, but this time it is compounded monthly<\/strong>.<\/li>\n<\/ul>\n\n\n\n

                          The Question:<\/strong> Does compounding monthly make enough of a difference for the 4.5% account to beat the 4.8% simple interest account?<\/p>\n\n\n\n

                          \n\n\n\n

                          Worked Solution<\/h3>\n\n\n\n

                          In an exam, you are taught to “adjust the variables” before plugging them into the formula A<\/mi>=<\/mo>P<\/mi>(<\/mo>1<\/mn>+<\/mo>r<\/mi>)<\/mo>n<\/mi><\/msup><\/mrow>A = P(1 + r)^n<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n

                          1. Adjust the Rate (<\/strong>r<\/mi>r<\/annotation><\/semantics><\/math>):<\/strong><\/p>\n\n\n\n

                          The annual rate is 4.5%. Since it’s monthly, we divide by 12.<\/p>\n\n\n\n

                          r<\/mi>=<\/mo>0.045<\/mn>\u00f7<\/mo>12<\/mn>=<\/mo>0.00375<\/mn><\/mrow>r = 0.045 \\div 12 = 0.00375<\/annotation><\/semantics><\/math><\/p>\n\n\n\n

                          2. Adjust the Periods (<\/strong>n<\/mi>n<\/annotation><\/semantics><\/math>):<\/strong><\/p>\n\n\n\n

                          The time is 3 years. Since it’s monthly, we multiply by 12.<\/p>\n\n\n\n

                          n<\/mi>=<\/mo>3<\/mn>\u00d7<\/mo>12<\/mn>=<\/mo>36<\/mn> periods<\/mtext><\/mrow>n = 3 \\times 12 = 36 \\text{ periods}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n

                          3. Apply the Formula:<\/strong><\/p>\n\n\n\n

                          A<\/mi>=<\/mo>4000<\/mn>(<\/mo>1<\/mn>+<\/mo>0.00375<\/mn>)<\/mo>36<\/mn><\/msup><\/mrow>A = 4000(1 + 0.00375)^{36}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n

                          A<\/mi>=<\/mo>4000<\/mn>(<\/mo>1.00375<\/mn>)<\/mo>36<\/mn><\/msup><\/mrow>A = 4000(1.00375)^{36}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n

                          A<\/mi>=<\/mo>4000<\/mn>\u00d7<\/mo>1.14424…<\/mn><\/mrow>A = 4000 \\times 1.14424…<\/annotation><\/semantics><\/math><\/p>\n\n\n\n

                          A<\/mi>=<\/mo>$<\/mi>4,576.99<\/mn><\/mrow>A = \\$4,576.99<\/annotation><\/semantics><\/math> (rounded to 2 decimal places)<\/p>\n\n\n\n
                          \n\n\n\n

                          Comparison Table: Which is better?<\/h3>\n\n\n\n
                          Investment Type<\/strong><\/td>Interest Rate<\/strong><\/td>Compounding<\/strong><\/td>Total after 3 years<\/strong><\/td><\/tr><\/thead>
                          Simple Interest<\/td>4.8% p.a.<\/td>N\/A<\/td>$4,576.00<\/strong><\/td><\/tr>
                          Compound Interest<\/td>4.5% p.a.<\/td>Annually<\/td>$4,564.66<\/strong><\/td><\/tr>
                          Compound Interest<\/strong><\/td>4.5% p.a.<\/strong><\/td>Monthly<\/strong><\/td>$4,576.99<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

                          The Verdict<\/h3>\n\n\n\n

                          By compounding monthly<\/strong>, the 4.5% account actually beats<\/strong> the 4.8% simple interest account\u2014but only by $0.99!<\/strong> Why this matters for your Year 12 studies:<\/strong><\/p>\n\n\n\n

                            \n
                          1. Precision:<\/strong> In Essential Maths, rounding too early can lose you marks. Always keep the full decimal in your calculator until the very final step.<\/li>\n\n\n\n
                          2. Comparison:<\/strong> QCAA often asks you to “Compare and Contrast.” A good answer would state: “While Account B has a lower nominal interest rate, the frequency of compounding (monthly) allows the interest to earn interest more often, eventually yielding a higher return than Account A.”<\/em><\/li>\n\n\n\n
                          3. The Formula Sheet:<\/strong> Remember, the QCAA formula sheet usually provides A<\/mi>=<\/mo>P<\/mi>(<\/mo>1<\/mn>+<\/mo>r<\/mi>)<\/mo>n<\/mi><\/msup><\/mrow>A = P(1+r)^n<\/annotation><\/semantics><\/math>. It is up to you<\/strong> to remember to divide the rate and multiply the time if it’s not annual.<\/li>\n<\/ol>\n\n\n\n

                            Would you like to move on to a practice problem for Unit 3 (like reading a Scale Drawing or House Plan), or would you like to see how credit card interest works?<\/strong><\/p>\n\n\n\n

                            <\/p>\n\n\n\n

                            <\/p>\n\n\n\n

                            <\/p>\n","protected":false},"excerpt":{"rendered":"

                            Unit 3: This moves us into Unit 3: Measurement, Scales, and Data. This unit is the bread and butter of the PSMT (Problem-Solving and Modelling Task), which is that big report you have to write in Year 12. In this topic, you aren’t just doing math; you\u2019re acting like a project manager or a builder. […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[8],"tags":[],"class_list":["post-1274","post","type-post","status-publish","format-standard","hentry","category-math"],"_links":{"self":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1274","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1274"}],"version-history":[{"count":2,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1274\/revisions"}],"predecessor-version":[{"id":1362,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1274\/revisions\/1362"}],"wp:attachment":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1274"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1274"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1274"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}