7:[["$","script",null,{"type":"application/ld+json","dangerouslySetInnerHTML":{"__html":"{\"@context\":\"https://schema.org\",\"@type\":\"Quiz\",\"name\":\"Quiz 16\",\"description\":\"SAT Math Advanced Quizzes - Quiz 16.\",\"url\":\"https://theschoolofmathematics.com/quiz/sat-math-advanced-quizzes-quiz-16\",\"educationalLevel\":\"Advanced\",\"numberOfQuestions\":10,\"isAccessibleForFree\":true,\"about\":{\"@type\":\"Thing\",\"name\":\"Advanced Quizzes\"}}"}}],["$","script",null,{"type":"application/ld+json","dangerouslySetInnerHTML":{"__html":"{\"@context\":\"https://schema.org\",\"@type\":\"BreadcrumbList\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https://theschoolofmathematics.com\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Quizzes\",\"item\":\"https://theschoolofmathematics.com/quiz\"},{\"@type\":\"ListItem\",\"position\":3,\"name\":\"Quiz 16\",\"item\":\"https://theschoolofmathematics.com/quiz/sat-math-advanced-quizzes-quiz-16\"}]}"}}],["$","div",null,{"className":"bg-gradient-to-b from-white via-gray-50 to-white","children":["$","div",null,{"className":"container mx-auto px-4 py-8 md:py-12","children":[["$","div",null,{"className":"mb-12","children":["$undefined",["$","div",null,{"className":"mb-8","children":["$","h1",null,{"className":"text-4xl md:text-5xl lg:text-6xl font-bold text-gray-900 mb-4","children":"Quiz 16"}]}],["$","div",null,{"className":"bg-white rounded-2xl shadow-lg p-6 md:p-8 border border-gray-200","children":[["$","div",null,{"className":"text-lg md:text-xl text-gray-700 mb-8 leading-relaxed prose prose-lg max-w-none","dangerouslySetInnerHTML":{"__html":"

SAT Math Advanced Quizzes - Quiz 16.

"}}],["$","div",null,{"className":"grid grid-cols-2 md:grid-cols-3 gap-4","children":["$undefined",["$","div",null,{"className":"flex items-center gap-3 p-4 bg-gray-50 rounded-xl border border-gray-200","children":[["$","div",null,{"className":"w-10 h-10 bg-gray-900 rounded-lg flex items-center justify-center flex-shrink-0","children":["$","svg",null,{"className":"w-5 h-5 text-white","fill":"none","stroke":"currentColor","viewBox":"0 0 24 24","children":["$","path",null,{"strokeLinecap":"round","strokeLinejoin":"round","strokeWidth":2,"d":"M8.228 9c.549-1.165 2.03-2 3.772-2 2.21 0 4 1.343 4 3 0 1.4-1.278 2.575-3.006 2.907-.542.104-.994.54-.994 1.093m0 3h.01M21 12a9 9 0 11-18 0 9 9 0 0118 0z"}]}]}],["$","div",null,{"children":[["$","div",null,{"className":"text-xs text-gray-500 font-medium","children":"Questions"}],["$","div",null,{"className":"text-sm font-bold text-gray-900","children":10}]]}]]}],["$","div",null,{"className":"flex items-center gap-3 p-4 bg-gray-50 rounded-xl border border-gray-200","children":[["$","div",null,{"className":"w-10 h-10 bg-gray-900 rounded-lg flex items-center justify-center flex-shrink-0","children":["$","svg",null,{"className":"w-5 h-5 text-white","fill":"none","stroke":"currentColor","viewBox":"0 0 24 24","children":["$","path",null,{"strokeLinecap":"round","strokeLinejoin":"round","strokeWidth":2,"d":"M9 12l2 2 4-4m6 2a9 9 0 11-18 0 9 9 0 0118 0z"}]}]}],["$","div",null,{"children":[["$","div",null,{"className":"text-xs text-gray-500 font-medium","children":"Level"}],["$","div",null,{"className":"text-sm font-bold text-gray-900","children":"Advanced"}]]}]]}],["$","div",null,{"className":"flex items-center gap-3 p-4 bg-gray-50 rounded-xl border border-gray-200","children":[["$","div",null,{"className":"w-10 h-10 bg-gray-900 rounded-lg flex items-center justify-center flex-shrink-0","children":["$","svg",null,{"className":"w-5 h-5 text-white","fill":"none","stroke":"currentColor","viewBox":"0 0 24 24","children":["$","path",null,{"strokeLinecap":"round","strokeLinejoin":"round","strokeWidth":2,"d":"M7 7h.01M7 3h5c.512 0 1.024.195 1.414.586l7 7a2 2 0 010 2.828l-7 7a2 2 0 01-2.828 0l-7-7A1.994 1.994 0 013 12V7a4 4 0 014-4z"}]}]}],["$","div",null,{"children":[["$","div",null,{"className":"text-xs text-gray-500 font-medium","children":"Category"}],["$","div",null,{"className":"text-sm font-bold text-gray-900","children":"Advanced Quizzes"}]]}]]}]]}]]}]]}],["$","$L17",null,{"quiz":{"id":2300,"prismaId":"cmq0jytzg0209jgodwzy23q1s","slug":"sat-math-advanced-quizzes-quiz-16","title":{"rendered":"Quiz 16"},"content":{"rendered":"

SAT Math Advanced Quizzes - Quiz 16.

"},"excerpt":{"rendered":"

SAT Math Advanced Quizzes - Quiz 16.

"},"featured_media":0,"featured_media_url":"$undefined","acf":{"duree_estimee":"$undefined","niveau_difficulte":"Advanced","categorie":"Advanced Quizzes","nombre_questions":10,"score_minimum":70,"ordre_questions":"Fixe","questions":[{"id":"cmq0jytzo020bjgodcdjcq7v3","texte_question":"

For all positive real numbers $$a$$ and $$b$$, the expression $$\\frac{a^2-b^2}{a^{4/3}+a^{2/3}b^{2/3}+b^{4/3}}$$ is equivalent to which of the following?

","type_question":"QCM","explication":"

Rewrite $$a^2-b^2$$ using powers of $$\\frac{2}{3}$$.

Let:

$$x=a^{2/3}$$

$$y=b^{2/3}$$

Then:

$$a^2=x^3$$

$$b^2=y^3$$

So, the numerator becomes:

$$x^3-y^3$$

Use the difference of cubes formula.

$$x^3-y^3=(x-y)(x^2+xy+y^2)$$

The denominator is: $$x^2+xy+y^2$$

Cancel the common factor.

$$x-y$$

Substitute back.

$$a^{2/3}-b^{2/3}$$

Final answer: $$a^{2/3}-b^{2/3}$$

","points":1,"temps_limite":120,"reponses":[{"texte":"

$$a^{2/3}-b^{2/3}$$

","correcte":true,"explication":"","imageUrl":"$undefined"},{"texte":"

$$a^{1/3}+b^{1/3}$$

","correcte":false,"explication":"","imageUrl":"$undefined"},{"texte":"

$$a^{2/3}+b^{2/3}$$

","correcte":false,"explication":"","imageUrl":"$undefined"},{"texte":"

$$a^{1/3}-b^{1/3}$$

","correcte":false,"explication":"","imageUrl":"$undefined"}]},{"id":"cmq0jytzz020hjgod0byn2h48","texte_question":"

A right circular cone has a height of $$15$$ units and a base radius of $$10$$ units. A plane parallel to the base cuts the cone at a point $$6$$ units above the base, forming a smaller cone and a frustum. What is the volume, in cubic units, of the frustum?

","type_question":"QCM","explication":"

The full cone has radius $$10$$ and height $$15$$. Its volume is:

$$V=\\frac{1}{3}\\pi r^2h$$

$$V=\\frac{1}{3}\\pi(10)^2(15)$$

$$V=500\\pi$$

The plane is $$6$$ units above the base, so the height of the smaller cone is:

$$15-6=9$$

By similarity, the radius of the smaller cone is:

$$\\frac{9}{15}\\cdot10=6$$

The volume of the smaller cone is:

$$V=\\frac{1}{3}\\pi(6)^2(9)$$

$$V=108\\pi$$

The frustum volume is the full cone volume minus the smaller cone volume.

$$500\\pi-108\\pi=392\\pi$$

Final answer: $$392\\pi$$

","points":1,"temps_limite":120,"reponses":[{"texte":"

$$30\\pi$$

","correcte":false,"explication":"","imageUrl":"$undefined"},{"texte":"

$$392\\pi$$

","correcte":true,"explication":"","imageUrl":"$undefined"},{"texte":"

$$150\\pi$$

","correcte":false,"explication":"","imageUrl":"$undefined"},{"texte":"

$$300\\pi$$

","correcte":false,"explication":"","imageUrl":"$undefined"}]},{"id":"cmq0jyu04020njgod6y86xf1m","texte_question":"

The system of equations $$y=x^2-8x+20$$ $$y=4x+k$$ has exactly one real solution, where $$k$$ is a constant. What is the value of $$k$$?

","type_question":"TexteLibre","explication":"

Set the two expressions for $$y$$ equal.

$$x^2-8x+20=4x+k$$

Move all terms to one side.

$$x^2-12x+20-k=0$$

For the system to have exactly one real solution, the quadratic must have discriminant $$0$$.

$$b^2-4ac=0$$

Here:

$$a=1$$, $$b=-12$$, $$c=20-k$$

Substitute.

$$(-12)^2-4(1)(20-k)=0$$

$$144-80+4k=0$$

$$64+4k=0$$

$$k=-16$$

Final answer: $$-16$$

","points":1,"temps_limite":120,"reponses":[{"texte":"

-16

","correcte":true,"explication":"","imageUrl":"$undefined"}]},{"id":"cmq0jyu08020qjgod103ccjfc","texte_question":"$18","type_question":"QCM","explication":"

The group is restricted to students who do not have a part-time job.

The total number of students who do not have a part-time job is: $$114$$

The number of those students who are not varsity athletes is: $$66$$

So, the probability is: $$\\frac{66}{114}$$

Simplify by dividing by $$6$$.

$$\\frac{66}{114}=\\frac{11}{19}$$

Final answer: $$\\frac{11}{19}$$

","points":1,"temps_limite":120,"reponses":[{"texte":"

$$\\frac{11}{19}$$

","correcte":true,"explication":"","imageUrl":"$undefined"},{"texte":"

$$\\frac{19}{40}$$

","correcte":false,"explication":"","imageUrl":"$undefined"},{"texte":"

$$\\frac{33}{57}$$

","correcte":false,"explication":"","imageUrl":"$undefined"},{"texte":"

$$\\frac{66}{240}$$

","correcte":false,"explication":"","imageUrl":"$undefined"}]},{"id":"cmq0jyu0d020wjgodgrnj7w4c","texte_question":"

The graph of the line $$ax+by=30$$ in the $$(x,y)$$-plane passes through the point $$(4,3)$$. If the $$x$$-intercept of the line is $$3$$ times its $$y$$-intercept, where $$a\\ne0$$ and $$b\\ne0$$, what is the value of $$b$$?

","type_question":"QCM","explication":"

For the line:

$$ax+by=30$$

The $$x$$-intercept occurs when $$y=0$$.

$$ax=30$$

$$x=\\frac{30}{a}$$

The $$y$$-intercept occurs when $$x=0$$.

$$by=30$$ $$y=\\frac{30}{b}$$

The $$x$$-intercept is $$3$$ times the $$y$$-intercept.

$$\\frac{30}{a}=3\\left(\\frac{30}{b}\\right)$$

$$\\frac{30}{a}=\\frac{90}{b}$$

Cross multiply.

$$30b=90a$$

$$b=3a$$

The line passes through $$(4,3)$$.

$$4a+3b=30$$

Substitute $$b=3a$$.

$$4a+3(3a)=30$$

$$13a=30$$

$$a=\\frac{30}{13}$$

Then:

$$b=3\\left(\\frac{30}{13}\\right)$$

$$b=\\frac{90}{13}$$

Final answer: $$\\frac{90}{13}$$

","points":1,"temps_limite":120,"reponses":[{"texte":"

$$3$$

","correcte":false,"explication":"","imageUrl":"$undefined"},{"texte":"

$$\\frac{15}{4}$$

","correcte":false,"explication":"","imageUrl":"$undefined"},{"texte":"

$$5$$

","correcte":false,"explication":"","imageUrl":"$undefined"},{"texte":"

$$\\frac{90}{13}$$

","correcte":true,"explication":"","imageUrl":"$undefined"}]},{"id":"cmq0jyu0i0212jgodb09xfa66","texte_question":"

The function $$f$$ is defined by $$f(x)=px^2+qx+r$$, where $$p$$, $$q$$, and $$r$$ are constants and $$0$$ < $$q$$ < $$12$$. The graph of $$y=f(x)$$ is a parabola such that $$f(-4)=f(8)$$. If $$q$$ is an integer, what is the greatest possible value of $$q-p$$?

","type_question":"QCM","explication":"

Since $$f(-4)=f(8)$$ the points with $$x=-4$$ and $$x=8$$ have the same $$y$$-value.

For a parabola, points with the same $$y$$-value are symmetric about the axis of symmetry.

So the axis of symmetry is halfway between $$-4$$ and $$8$$.

$$x=\\frac{-4+8}{2}$$

$$x=2$$

For the quadratic function $$f(x)=px^2+qx+r$$ the axis of symmetry is:

$$x=-\\frac{q}{2p}$$

So:

$$-\\frac{q}{2p}=2$$

Multiply both sides by $$2p$$.

$$-q=4p$$

So:

$$p=-\\frac{q}{4}$$

Now find $$q-p$$.

$$q-p=q-\\left(-\\frac{q}{4}\\right)$$

$$q-p=\\frac{5q}{4}$$

Since $$q$$ is an integer and $$0$$ < $$q$$ < $$12$$ the greatest possible value of $$q$$ is: $$11$$

Therefore:

$$q-p=\\frac{5(11)}{4}$$

$$q-p=\\frac{55}{4}$$

Final answer: $$\\frac{55}{4}$$

","points":1,"temps_limite":120,"reponses":[{"texte":"

$$\\frac{11}{4}$$

","correcte":false,"explication":"","imageUrl":"$undefined"},{"texte":"

$$\\frac{1}{2}$$

","correcte":false,"explication":"","imageUrl":"$undefined"},{"texte":"

$$\\frac{55}{4}$$

","correcte":true,"explication":"","imageUrl":"$undefined"},{"texte":"

$$\\frac{1}{4}$$

","correcte":false,"explication":"","imageUrl":"$undefined"}]},{"id":"cmq0jyu0n0218jgodt29m0ien","texte_question":"

In $$\\triangle XYZ$$, point $$M$$ lies on $$\\overline{YZ}$$ such that $$XM$$ bisects $$\\angle YXZ$$. If $$MN\\parallel XY$$ and $$MP\\parallel XZ$$, where $$N$$ is on $$\\overline{XZ}$$ and $$P$$ is on $$\\overline{XY}$$, and $$\\angle YXZ=80^\\circ$$ and $$\\angle XYZ=50^\\circ$$, what is the measure of $$\\angle NMP$$?

","type_question":"QCM","explication":"

Since $$\\angle YXZ=80^\\circ$$ and $$XM$$ bisects $$\\angle YXZ$$, each half is:

$$\\frac{80^\\circ}{2}=40^\\circ$$

The given angle $$\\angle XYZ=50^\\circ$$.

Since the angles of a triangle add to $$180^\\circ$$:

$$\\angle XZY=180^\\circ-80^\\circ-50^\\circ$$

$$\\angle XZY=50^\\circ$$

Because $$MN\\parallel XY$$ and $$MP\\parallel XZ$$, the angle between $$MN$$ and $$MP$$ is equal to the angle between $$XY$$ and $$XZ$$. That angle is $$\\angle YXZ$$.

So:

$$\\angle NMP=80^\\circ$$

Final answer: $$80^\\circ$$

","points":1,"temps_limite":120,"reponses":[{"texte":"

$$40^\\circ$$

","correcte":false,"explication":"","imageUrl":"$undefined"},{"texte":"

$$50^\\circ$$

","correcte":false,"explication":"","imageUrl":"$undefined"},{"texte":"

$$80^\\circ$$

","correcte":true,"explication":"","imageUrl":"$undefined"},{"texte":"

$$100^\\circ$$

","correcte":false,"explication":"","imageUrl":"$undefined"}]},{"id":"cmq0jyu0r021ejgodu4odm6tp","texte_question":"

A circle lies entirely in the first quadrant of the $$(x,y)$$-plane and is tangent to both axes. The circle is also tangent to the line $$5x+12y=180$$ What is the sum of all possible values for the radius of the circle?

","type_question":"QCM","explication":"

Since the circle is tangent to both axes and lies in the first quadrant, its center is: $$(r,r)$$ where $$r$$ is the radius.

The distance from $$(r,r)$$ to the line $$5x+12y=180$$ must equal $$r$$.

Rewrite the line.

$$5x+12y-180=0$$

Use the point-to-line distance formula.

$$\\frac{|5r+12r-180|}{\\sqrt{5^2+12^2}}=r$$

$$\\frac{|17r-180|}{13}=r$$

So:

$$|17r-180|=13r$$

Case 1:

$$17r-180=13r$$

$$4r=180$$

$$r=45$$

Case 2:

$$180-17r=13r$$

$$180=30r$$

$$r=6$$

The sum of all possible radii is:

$$45+6=51$$

Final answer: $$51$$

","points":1,"temps_limite":120,"reponses":[{"texte":"

$$12$$

","correcte":false,"explication":"","imageUrl":"$undefined"},{"texte":"

$$24$$

","correcte":false,"explication":"","imageUrl":"$undefined"},{"texte":"

$$36$$

","correcte":false,"explication":"","imageUrl":"$undefined"},{"texte":"

$$51$$

","correcte":true,"explication":"","imageUrl":"$undefined"}]},{"id":"cmq0jyu0w021kjgodmh3o0poa","texte_question":"

The numbers $$a$$, $$b$$, $$c$$, $$m$$, and $$n$$ are positive constants, where $$b$$ > $$c$$. The number $$a$$ is $$500\\%$$ greater than the positive difference between $$b$$ and $$c$$, and $$c$$ is $$80\\%$$ of $$b$$. If $$a$$ is $$m\\%$$ greater than $$c$$, and $$c$$ is $$n\\%$$ of $$a$$, what is the product of $$m$$ and $$n$$? Round your answer to the nearest integer.

","type_question":"TexteLibre","explication":"

Since $$c$$ is $$80\\%$$ of $$b$$: $$c=0.8b$$

The positive difference between $$b$$ and $$c$$ is:

$$b-c=b-0.8b$$

$$b-c=0.2b$$

The number $$a$$ is $$500\\%$$ greater than this difference.

That means $$a$$ is $$6$$ times the difference.

$$a=6(0.2b)$$

$$a=1.2b$$

Now compare $$a$$ to $$c$$.

$$a=1.2b$$

$$c=0.8b$$

The percent by which $$a$$ is greater than $$c$$ is:

$$m=\\frac{a-c}{c}\\cdot100$$

$$m=\\frac{1.2b-0.8b}{0.8b}\\cdot100$$

$$m=50$$

Now find $$n$$, where $$c$$ is $$n\\%$$ of $$a$$.

$$n=\\frac{c}{a}\\cdot100$$

$$n=\\frac{0.8b}{1.2b}\\cdot100$$

$$n=66.666\\ldots$$

Now multiply.

$$mn=50(66.666\\ldots)$$

$$mn=3333.333\\ldots$$

Rounded to the nearest integer: $$3333$$

Final answer: $$3333$$

","points":1,"temps_limite":120,"reponses":[{"texte":"

3333

","correcte":true,"explication":"","imageUrl":"$undefined"}]},{"id":"cmq0jyu10021njgodn8dce83r","texte_question":"

In $$\\triangle ABC$$, $$BD$$ is an angle bisector of $$\\angle ABC$$, with point $$D$$ on side $$AC$$. A line is drawn through $$D$$ parallel to $$BC$$, intersecting side $$AB$$ at point $$E$$. If $$\\angle ABD=35^\\circ$$ and $$\\angle BDC=115^\\circ$$, what is the measure of $$\\angle AED$$?

","type_question":"QCM","explication":"

Since $$BD$$ bisects $$\\angle ABC$$: $$\\angle ABD=\\angle DBC$$

Given:

$$\\angle ABD=35^\\circ$$

So:

$$\\angle ABC=35^\\circ+35^\\circ$$

$$\\angle ABC=70^\\circ$$

Since $$D$$ lies on $$AC$$, angles $$BDC$$ and $$BDA$$ are supplementary.

$$\\angle BDA=180^\\circ-115^\\circ$$

$$\\angle BDA=65^\\circ$$

In triangle $$ABD$$:

$$\\angle BAD=180^\\circ-35^\\circ-65^\\circ$$

$$\\angle BAD=80^\\circ$$

Since $$AD$$ lies on $$AC$$, $$\\angle BAC=80^\\circ$$.

Now find $$\\angle ACB$$ in triangle $$ABC$$.

$$\\angle ACB=180^\\circ-80^\\circ-70^\\circ$$

$$\\angle ACB=30^\\circ$$

Because $$ED\\parallel BC$$, angle $$AED$$ corresponds to angle $$ABC$$.

So:

$$\\angle AED=70^\\circ$$

Final answer: $$70^\\circ$$

","points":1,"temps_limite":120,"reponses":[{"texte":"

$$65^\\circ$$

","correcte":false,"explication":"","imageUrl":"$undefined"},{"texte":"

$$70^\\circ$$

","correcte":true,"explication":"","imageUrl":"$undefined"},{"texte":"

$$80^\\circ$$

","correcte":false,"explication":"","imageUrl":"$undefined"},{"texte":"

$$115^\\circ$$

","correcte":false,"explication":"","imageUrl":"$undefined"}]}]},"categories":["Advanced Quizzes"],"date":"2026-06-05T06:38:08.717Z","modified":"2026-06-05T06:38:08.717Z"}}],["$","div",null,{"className":"container mx-auto px-4 pb-12","children":["$","$L19",null,{"targetType":"quiz","targetSlug":"sat-math-advanced-quizzes-quiz-16"}]}]]}]}]]

Quiz 16

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