{"id":1826,"date":"2026-03-17T13:22:41","date_gmt":"2026-03-17T13:22:41","guid":{"rendered":"https:\/\/www.think10x.ai\/?p=1826"},"modified":"2026-03-25T00:48:49","modified_gmt":"2026-03-25T00:48:49","slug":"percent-decrease-in-an-exponential-function","status":"publish","type":"post","link":"https:\/\/www.think10x.ai\/blog\/percent-decrease-in-an-exponential-function\/","title":{"rendered":"Exponential Function Problem: How to Find Percent Decrease Step-by-Step?"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"1826\" class=\"elementor elementor-1826\" data-elementor-post-type=\"post\">\n\t\t\t\t<div class=\"elementor-element elementor-element-f423cf0 e-flex e-con-boxed e-con e-parent\" data-id=\"f423cf0\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-92a0413 elementor-widget elementor-widget-text-editor\" data-id=\"92a0413\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<h2>The Question<\/h2><p><span style=\"font-weight: 400;\">The function f is defined by <strong>f(x)<\/strong> = 78(0.39)<sup>x<\/sup>. For any positive integer n, f(n) is <strong>p%<\/strong> less than <strong>f(n &#8211; 1)<\/strong>. What is the value of <strong>p<\/strong>?<\/span><\/p><p><span style=\"font-weight: 400;\">(A) 78<\/span><\/p><p><span style=\"font-weight: 400;\">(B) 61 <strong>\u2713<\/strong><\/span><\/p><p><span style=\"font-weight: 400;\">(C) 39<\/span><\/p><p><span style=\"font-weight: 400;\">(D) 22<\/span><\/p><h2 class=\"text-text-100 mt-3 -mb-1 text-[1.125rem] font-bold\">Why Most Students Get This Wrong<\/h2><p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">This is a classic SAT trap. You see the base <strong>0.39<\/strong> and assume the percent decrease is <strong>39%<\/strong>. That feels right, but it is wrong.<\/p><p>An exponential function of the form f(x) = <strong>a(b)<sup>x<\/sup><\/strong> has two key parts. The coefficient a is the starting value, and the base b controls how the function changes at each step. In this function, b is always the decay factor, meaning the proportion that remains, not the proportion that is lost.<\/p><p>The base does not tell you what is lost. It tells you what stays.<\/p><div class=\"flex flex-col text-sm pb-25\"><section class=\"text-token-text-primary w-full focus:outline-none [--shadow-height:45px] has-data-writing-block:pointer-events-none has-data-writing-block:-mt-(--shadow-height) has-data-writing-block:pt-(--shadow-height) [&amp;:has([data-writing-block])&gt;*]:pointer-events-auto scroll-mt-[calc(var(--header-height)+min(200px,max(70px,20svh)))]\" dir=\"auto\" data-turn-id=\"request-WEB:f7872cfd-c600-416d-890c-519cda6d2705-3\" data-testid=\"conversation-turn-8\" data-scroll-anchor=\"true\" data-turn=\"assistant\"><div class=\"text-base my-auto mx-auto pb-10 [--thread-content-margin:var(--thread-content-margin-xs,calc(var(--spacing)*4))] @w-sm\/main:[--thread-content-margin:var(--thread-content-margin-sm,calc(var(--spacing)*6))] @w-lg\/main:[--thread-content-margin:var(--thread-content-margin-lg,calc(var(--spacing)*16))] px-(--thread-content-margin)\"><div class=\"[--thread-content-max-width:40rem] @w-lg\/main:[--thread-content-max-width:48rem] mx-auto max-w-(--thread-content-max-width) flex-1 group\/turn-messages focus-visible:outline-hidden relative flex w-full min-w-0 flex-col agent-turn\"><div class=\"flex max-w-full flex-col gap-4 grow\"><div class=\"min-h-8 text-message relative flex w-full flex-col items-end gap-2 text-start break-words whitespace-normal outline-none keyboard-focused:focus-ring [.text-message+&amp;]:mt-1\" dir=\"auto\" tabindex=\"0\" data-message-author-role=\"assistant\" data-message-id=\"027bcb96-f87f-4123-9302-294c04a3e3af\" data-message-model-slug=\"gpt-5-3\" data-turn-start-message=\"true\"><div class=\"flex w-full flex-col gap-1 empty:hidden\"><div class=\"markdown prose dark:prose-invert w-full wrap-break-word dark markdown-new-styling\"><h2 class=\"text-text-100 mt-3 -mb-1 text-[1.125rem] font-bold\">Quick Answer<\/h2><p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">p = <strong>61<\/strong>. The correct answer is B.<\/p><p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The base 0.39 means 39% remains each step. The percent lost is 100% &#8211; 39% = <strong>61%<\/strong>.<\/p><h3 class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">General Rule<\/h3><p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">In any function of the form f(x) =<strong>a(b)<sup>x <\/sup><\/strong>where 0 &lt; b &lt; 1,<\/p><ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\"><li class=\"whitespace-normal break-words pl-2\"><strong>b<\/strong> is the <strong>fraction<\/strong> that remains each step<\/li><li class=\"whitespace-normal break-words pl-2\">Percent decrease = (<strong>1 &#8211; b<\/strong>) <strong>x<\/strong> <strong>100%<\/strong><\/li><li class=\"whitespace-normal break-words pl-2\">Applied here, (1 &#8211; 0.39) x 100% = <strong>61%<\/strong><\/li><\/ul><h2>Video Walkthrough<\/h2><p>We created a step-by-step video walkthrough of this <strong>exponential function SAT problem<\/strong>. Watch how we write out two consecutive terms, form a ratio to find the decay factor, and then back-solve to find the percent decrease.<\/p><\/div><\/div><\/div><\/div><\/div><\/div><\/section><\/div>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-97d272b e-flex e-con-boxed e-con e-parent\" data-id=\"97d272b\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-8f872ce elementor-widget elementor-widget-video\" data-id=\"8f872ce\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;youtube_url&quot;:&quot;https:\\\/\\\/youtu.be\\\/O3zyR51xWfQ?si=j2YGKvr3zz0vyo8n&quot;,&quot;video_type&quot;:&quot;youtube&quot;,&quot;controls&quot;:&quot;yes&quot;}\" data-widget_type=\"video.default\">\n\t\t\t\t\t\t\t<div class=\"elementor-wrapper elementor-open-inline\">\n\t\t\t<div class=\"elementor-video\"><\/div>\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-156e645 e-flex e-con-boxed e-con e-parent\" data-id=\"156e645\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-6c085ae elementor-widget elementor-widget-text-editor\" data-id=\"6c085ae\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<h2 class=\"text-text-100 mt-3 -mb-1 text-[1.125rem] font-bold\">Step-by-Step Solution<\/h2><p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Here is the full algebraic proof, useful if you want to show your work or understand why the shortcut works.<\/p><h3 class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Step 1. Write out two back-to-back terms<\/h3><p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The problem says f(n) is p% less than <strong>f(n &#8211; 1)<\/strong>. Start by writing both terms explicitly.<\/p><p class=\"font-claude-response-body break-words whitespace-pre-wrap leading-[1.7]\">f(n) = <strong>78(0.39)<sup>n <\/sup><\/strong>and f(n &#8211; 1) = <strong>78(0.39)<sup>n-1<\/sup><\/strong><\/p><p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">These are just two consecutive outputs of the same function, one step apart.<\/p><h3 class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Step 2. Divide to find what fraction one term is of the previous<\/h3><p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Form a ratio by dividing the newer term by the older one.<\/p><p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">f(n) \/ f(n &#8211; 1) = [<strong>78(0.39)<sup>n<\/sup><\/strong>] \/ [<strong>78(0.39)<sup>n-1<\/sup><\/strong>]<\/p><p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The 78 cancels top and bottom. Then <strong>0.39<sup>n<\/sup><\/strong> \/ <strong>0.39)<sup>n-1<\/sup><\/strong> = <strong>0.39<sup>1<\/sup><\/strong> = 0.39.<\/p><p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">So f(n) \/ f(n &#8211; 1) = <strong>0.39<\/strong><\/p><p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Every term is exactly 0.39x the one before it.<\/p><h3 class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Step 3. Translate &#8220;p% less&#8221; into a math equation<\/h3><p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">If something is p% less, it keeps (100 &#8211; p)% of what it was. In decimal form, that means multiplying by <strong>(1 &#8211; p\/100)<\/strong>.<\/p><p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">f(n) = <strong>(1 &#8211; p\/100)<\/strong> x <strong>f(n &#8211; 1)<\/strong><\/p><p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Divide both sides by f(n &#8211; 1), f(n) \/ f(n &#8211; 1) = <strong>1 &#8211; p\/100<\/strong><\/p><p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A 20% decrease means you keep 80%, so you multiply by 0.80 = 1 &#8211; 20\/100. Same logic here.<\/p><h3 class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Step 4. Set the two expressions equal and solve for p<\/h3><p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">From Step 2, the ratio = <strong>0.39<\/strong>. From Step 3, the ratio = <strong>1 &#8211; p\/100<\/strong>. They describe the same thing, so set them equal.<\/p><ul><li>0.39 = 1 &#8211; p\/100<\/li><li>p\/100 = 1 &#8211; 0.39 = 0.61<\/li><li>p = 0.61 x 100 = 61<\/li><li>p = <strong>61<\/strong><\/li><\/ul><h2 class=\"text-text-100 mt-3 -mb-1 text-[1.125rem] font-bold\">Fast SAT Shortcut<\/h2><p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">You do not need the full ratio method once you recognize the pattern.<\/p><p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">In <strong>a(b)<sup>x <\/sup><\/strong>where 0 &lt; b &lt; 1, subtract the base from 1 and convert to a percent.<\/p><p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Base is 0.39, so 1 &#8211; 0.39 = 0.61, which means a 61% decrease.<\/p><h2 class=\"text-text-100 mt-3 -mb-1 text-[1.125rem] font-bold\">Why Each Wrong Answer Looks Tempting<\/h2><p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>A is 78.<\/strong> This is the starting coefficient, not the rate of change. When you form the ratio f(n)\/f(n-1), the 78 cancels out completely. It has no effect on p.<\/p><p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>B is 61.<\/strong> Correct. The base 0.39 tells you 39% remains. The percent lost is 100 &#8211; 39 = 61.<\/p><p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>C is 39.<\/strong> The most common mistake. Students read the base 0.39 and treat it as the percent decrease. But 0.39 is the percent that stays, not the percent that disappears.<\/p><p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>D is 22.<\/strong> Comes from subtracting 39 from 61, or incorrectly mixing the coefficient and base. There is no clean mathematical path to 22 from this problem. If you got this, retrace your algebra from Step 2.<\/p><h2>Build the Pattern With These Examples<\/h2>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-65bfc89 eael-table-align-center eael-dt-th-align-left elementor-widget elementor-widget-eael-data-table\" data-id=\"65bfc89\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"eael-data-table.default\">\n\t\t\t\t\t\t\t<div class=\"eael-data-table-wrap\" data-table_id=\"65bfc89\" id=\"eael-data-table-wrapper-65bfc89\" data-custom_responsive=\"false\">\n\t\t\t<table class=\"tablesorter eael-data-table center\" id=\"eael-data-table-65bfc89\">\n\t\t\t    <thead>\n\t\t\t        <tr class=\"table-header\">\n\t\t\t\t\t\t\t\t\t            <th class=\"\" id=\"\" colspan=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"data-table-header-text\">Base (b)<\/span><\/th>\n\t\t\t        \t\t\t\t            <th class=\"\" id=\"\" colspan=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"data-table-header-text\">What it means<\/span><\/th>\n\t\t\t        \t\t\t\t            <th class=\"\" id=\"\" colspan=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"data-table-header-text\">Percent change<\/span><\/th>\n\t\t\t        \t\t\t\t        <\/tr>\n\t\t\t    <\/thead>\n\t\t\t  \t<tbody>\n\t\t\t\t\t\t\t\t\t\t\t<tr>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t0.39\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t39% remains each step\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t61% decrease \u2190 this problem\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/tr>\n\t\t\t        \t\t\t\t\t\t<tr>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t0.82\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t82% remains each step\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t18% decrease\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/tr>\n\t\t\t        \t\t\t\t\t\t<tr>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t0.50\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t50% remains each step\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t50% decrease\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/tr>\n\t\t\t        \t\t\t\t\t\t<tr>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t1.12\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\tGrows by 12% each step\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t   \t\t\t\t\t\t\t\t\t\t\t<td colspan=\"\" rowspan=\"\" class=\"\" id=\"\">\n\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"td-content-wrapper\"><div class=\"td-content\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t12% increase\t\t\t\t\t\t\t\t\t\t\t\t<\/div><\/div>\n\t\t\t\t\t\t\t\t\t\t\t<\/td>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/tr>\n\t\t\t        \t\t\t    <\/tbody>\n\t\t\t<\/table>\n\t\t<\/div>\n\t  \t\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-dfb4972 elementor-widget elementor-widget-text-editor\" data-id=\"dfb4972\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<h2>Quick Transfer Check<\/h2><p><span style=\"font-weight: 400;\">If the base were <\/span><b>0.82<\/b><span style=\"font-weight: 400;\">, the percent decrease would be <\/span><b>(1 &#8211; 0.82) \u00d7 100% = 18%<\/b><span style=\"font-weight: 400;\">. Same rule, different base.<\/span><\/p><h2>Frequently Asked Questions<\/h2>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-13c16ae e-flex e-con-boxed e-con e-parent\" data-id=\"13c16ae\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-61a87a8 elementor-widget elementor-widget-eael-adv-accordion\" data-id=\"61a87a8\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"eael-adv-accordion.default\">\n\t\t\t\t\t            <div class=\"eael-adv-accordion\" id=\"eael-adv-accordion-61a87a8\" data-scroll-on-click=\"no\" data-scroll-speed=\"300\" data-accordion-id=\"61a87a8\" data-accordion-type=\"accordion\" data-toogle-speed=\"300\">\n            <div class=\"eael-accordion-list\">\n\t\t\t\t\t<div id=\"why-is-the-answer-61-and-not-39\" class=\"elementor-tab-title eael-accordion-header\" tabindex=\"0\" data-tab=\"1\" aria-controls=\"elementor-tab-content-1021\"><span class=\"eael-advanced-accordion-icon-closed\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-advanced-accordion-icon-opened\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-accordion-tab-title\">Why is the answer 61 and not 39?<\/span><svg aria-hidden=\"true\" class=\"fa-toggle e-font-icon-svg e-fas-angle-right\" viewBox=\"0 0 256 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"><\/path><\/svg><\/div><div id=\"elementor-tab-content-1021\" class=\"eael-accordion-content clearfix\" data-tab=\"1\" aria-labelledby=\"why-is-the-answer-61-and-not-39\"><p><span style=\"font-weight: 400\">Because 0.39 is the fraction that remains after each step, not the fraction that disappears. The percent decrease is 1 &#8211; 0.39 = 0.61, or 61%.<\/span><\/p><\/div>\n\t\t\t\t\t<\/div><div class=\"eael-accordion-list\">\n\t\t\t\t\t<div id=\"does-the-base-represent-percent-decrease-or-percent-remaining\" class=\"elementor-tab-title eael-accordion-header\" tabindex=\"0\" data-tab=\"2\" aria-controls=\"elementor-tab-content-1022\"><span class=\"eael-advanced-accordion-icon-closed\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-advanced-accordion-icon-opened\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-accordion-tab-title\">Does the base represent percent decrease or percent remaining?<\/span><svg aria-hidden=\"true\" class=\"fa-toggle e-font-icon-svg e-fas-angle-right\" viewBox=\"0 0 256 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"><\/path><\/svg><\/div><div id=\"elementor-tab-content-1022\" class=\"eael-accordion-content clearfix\" data-tab=\"2\" aria-labelledby=\"does-the-base-represent-percent-decrease-or-percent-remaining\"><p><span style=\"font-weight: 400\">The base represents the percent remaining. To find the percent decrease, subtract the base from 1 and multiply by 100.<\/span><\/p><\/div>\n\t\t\t\t\t<\/div><div class=\"eael-accordion-list\">\n\t\t\t\t\t<div id=\"what-formula-converts-a-decay-factor-into-a-percent-decrease\" class=\"elementor-tab-title eael-accordion-header\" tabindex=\"0\" data-tab=\"3\" aria-controls=\"elementor-tab-content-1023\"><span class=\"eael-advanced-accordion-icon-closed\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-advanced-accordion-icon-opened\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-accordion-tab-title\">What formula converts a decay factor into a percent decrease?<\/span><svg aria-hidden=\"true\" class=\"fa-toggle e-font-icon-svg e-fas-angle-right\" viewBox=\"0 0 256 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"><\/path><\/svg><\/div><div id=\"elementor-tab-content-1023\" class=\"eael-accordion-content clearfix\" data-tab=\"3\" aria-labelledby=\"what-formula-converts-a-decay-factor-into-a-percent-decrease\"><p><span style=\"font-weight: 400\">If the base is b and 0 &lt; b &lt; 1, then percent decrease = (1 \u2212 b) \u00d7 100%.<\/span><\/p><\/div>\n\t\t\t\t\t<\/div><div class=\"eael-accordion-list\">\n\t\t\t\t\t<div id=\"why-does-the-coefficient-78-not-affect-p\" class=\"elementor-tab-title eael-accordion-header\" tabindex=\"0\" data-tab=\"4\" aria-controls=\"elementor-tab-content-1024\"><span class=\"eael-advanced-accordion-icon-closed\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-advanced-accordion-icon-opened\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-accordion-tab-title\">Why does the coefficient 78 not affect p?<\/span><svg aria-hidden=\"true\" class=\"fa-toggle e-font-icon-svg e-fas-angle-right\" viewBox=\"0 0 256 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"><\/path><\/svg><\/div><div id=\"elementor-tab-content-1024\" class=\"eael-accordion-content clearfix\" data-tab=\"4\" aria-labelledby=\"why-does-the-coefficient-78-not-affect-p\"><p><span style=\"font-weight: 400\">Because when you compare consecutive terms using a ratio, the 78 cancels out. The percent change depends only on the base.<\/span><\/p><\/div>\n\t\t\t\t\t<\/div><div class=\"eael-accordion-list\">\n\t\t\t\t\t<div id=\"if-the-base-is-039-why-does-that-not-mean-a-39-decrease\" class=\"elementor-tab-title eael-accordion-header\" tabindex=\"0\" data-tab=\"5\" aria-controls=\"elementor-tab-content-1025\"><span class=\"eael-advanced-accordion-icon-closed\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-advanced-accordion-icon-opened\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-accordion-tab-title\">If the base is 0.39, why does that not mean a 39% decrease?<\/span><svg aria-hidden=\"true\" class=\"fa-toggle e-font-icon-svg e-fas-angle-right\" viewBox=\"0 0 256 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"><\/path><\/svg><\/div><div id=\"elementor-tab-content-1025\" class=\"eael-accordion-content clearfix\" data-tab=\"5\" aria-labelledby=\"if-the-base-is-039-why-does-that-not-mean-a-39-decrease\"><p><span style=\"font-weight: 400\">A 39% decrease would mean 61% remains. But here the base is 0.39, which means only 39% remains. So the decrease must be 61%.<\/span><\/p><\/div>\n\t\t\t\t\t<\/div><div class=\"eael-accordion-list\">\n\t\t\t\t\t<div id=\"how-is-this-different-from-exponential-growth\" class=\"elementor-tab-title eael-accordion-header\" tabindex=\"0\" data-tab=\"6\" aria-controls=\"elementor-tab-content-1026\"><span class=\"eael-advanced-accordion-icon-closed\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-advanced-accordion-icon-opened\"><svg aria-hidden=\"true\" class=\"fa-accordion-icon e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span><span class=\"eael-accordion-tab-title\">How is this different from exponential growth?<\/span><svg aria-hidden=\"true\" class=\"fa-toggle e-font-icon-svg e-fas-angle-right\" viewBox=\"0 0 256 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"><\/path><\/svg><\/div><div id=\"elementor-tab-content-1026\" class=\"eael-accordion-content clearfix\" data-tab=\"6\" aria-labelledby=\"how-is-this-different-from-exponential-growth\"><p><span style=\"font-weight: 400\">In exponential growth, the base is greater than 1, so the values increase each step. In exponential decay, the base is between 0 and 1, so the values decrease.<\/span><\/p><\/div>\n\t\t\t\t\t<\/div><\/div>\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-b0a530b e-flex e-con-boxed e-con e-parent\" data-id=\"b0a530b\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-9e85bfa elementor-widget elementor-widget-text-editor\" data-id=\"9e85bfa\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<h2>Want a Step-by-Step Video for Your Own Exponential Problem?<\/h2><p>The above example\u2019s video explanation was generated using<a href=\"https:\/\/www.think10x.ai\/\">\u00a0Think10x.ai<\/a>.<\/p><p>Upload a<a href=\"https:\/\/www.think10x.ai\/take-a-clear-photo-of-a-question\/\">\u00a0clear photo<\/a> of any math problem, including e<span style=\"font-weight: 400;\">xponential equations, exponential decay problems, algebra, geometry, or calculus, and our tool will turn it into <\/span>narrated,<a href=\"https:\/\/www.think10x.ai\/learn-from-video-explanations\/\">\u00a0animated explanation<\/a>\u00a0in minutes.<\/p><p><img decoding=\"async\" class=\"emoji\" role=\"img\" draggable=\"false\" src=\"https:\/\/s.w.org\/images\/core\/emoji\/17.0.2\/svg\/1f449.svg\" alt=\"\ud83d\udc49\" \/>\u00a0Try it at<a href=\"https:\/\/www.think10x.ai\/create\/\">\u00a0<b>Think10x.ai<\/b><\/a><\/p><p>Private by default.<a href=\"https:\/\/www.think10x.ai\/captions-and-transcripts-in-educational-videos\/\">\u00a0Captions<\/a>\u00a0included. Built for\u00a0<a href=\"https:\/\/mentomind.ai\/\" target=\"_blank\" rel=\"noopener\">tutors<\/a>\u00a0and\u00a0<a href=\"https:\/\/mentomind.ai\/for-students\/\" target=\"_blank\" rel=\"noopener\">students<\/a>.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>The Question The function f is defined by f(x) = 78(0.39)x. For any positive integer n, f(n) is p% less than f(n &#8211; 1). What is the value of p? (A) 78 (B) 61 \u2713 (C) 39 (D) 22 Why Most Students Get This Wrong This is a classic SAT trap. You see the base [&hellip;]<\/p>\n","protected":false},"author":3,"featured_media":1884,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"fifu_image_url":"https:\/\/media.mentomind.ai\/img\/t10x\/bp\/Exponential_Function_Problem.webp","fifu_image_alt":"Exponential function percent decrease problem with step-by-step solution using Think10X image-to-video explanation tool","footnotes":""},"categories":[8,1],"tags":[],"class_list":["post-1826","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-blogs","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/posts\/1826","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/comments?post=1826"}],"version-history":[{"count":5,"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/posts\/1826\/revisions"}],"predecessor-version":[{"id":1883,"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/posts\/1826\/revisions\/1883"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/media\/1884"}],"wp:attachment":[{"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/media?parent=1826"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/categories?post=1826"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.think10x.ai\/blog\/wp-json\/wp\/v2\/tags?post=1826"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}